moment maps and riemannian symmetric pairs we study hamiltonian actions of a compact lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold the fixed point set of the involution on the manifold is a lagrangian submanifold we investigate its image under the moment map and conclude that the intersection with the weyl chamber is an easily described subpolytope of the kirwan polytope of special interest is the integral kahler case where much stronger results hold in particular we obtain convexity theorems for closures of orbits of the noncompact dual group in the sense of the theory of symmetric pairs in the abelian case these results were obtained earlier by duistermaat we derive explicit inequalities for polytopes associated with real flag varieties

a note on observability of boolean control networks abstract the observability of boolean control networks is investigated the pairs of states are classified into three classes i diagonal ii h distinguishable and iii h indistinguishable for h indistinguishable pairs we construct a matrix w called the transferable matrix which indicates the control transferability among h indistinguishable pairs modifying w yields a boolean matrix u 0 which is used as the initial matrix for an iterative algorithm after finite iterations a stable u is reached which is called the observability matrix it is proved that a boolean control network is observable if and only if the last column of u col r 1 u 1 r some numerical examples are presented

output feedback stabilization control design for boolean control networks this paper investigates the output feedback stabilization of boolean control networks bcns by using the semi tensor product method and presents a number of new results first based on the algebraic expression of bcns a necessary and sufficient condition is presented for the existence of output feedback stabilizers second a constructive procedure is proposed to design output feedback stabilization controllers for bcns the study of an illustrative example shows that the new results obtained in this paper are very effective in designing output feedback stabilizers for bcns

t ideals and superazumaya algebras equivalent to being azumaya of finite rank thus one would expectazumaya algebras to play a role in the theory of t prime algebras a pointof view already taken in 2 wxour goal in this paper is to link the theory of t prime algebras to theseother important theories the theory of zr2z graded polynomial identi ties and the graded theory of azumaya algebras following kemer we believe that there is a natural connection between pi theory and zr2z graded algebras and that pi theorems should tend to have natural zr2z graded analogues our main result will be such an analogue of the artin procesi theorem theorems 13 18 in the process we consider the brauer wall group defined by c t c wall in 17 and further developed by h bass in 1 and c small in 16 cf wx wx wxwx8 for further results we will construct a pi criterion for a zr2z gradedalgebra to be zr2z graded projective of constant rank throughout thispaper all algebras will be with 1 over an infinite field

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

number and length of attractors in a critical kauffman model with connectivity one the kauffman model describes a system of randomly connected nodes with dynamics based on boolean update functions though it is a simple model it exhibits very complex behavior for critical parameter values at the boundary between a frozen and a disordered phase and is therefore used for studies of real network problems we prove here that the mean number and mean length of attractors in critical random boolean networks with connectivity one both increase faster than any power law with network size we derive these results by generating the networks through a growth process and by calculating lower bounds

brief paper controllability of boolean control networks with time delays in states this paper investigates the controllability of boolean networks with time invariant delays in states after a brief introduction on converting the logic dynamics to discrete time delay systems the controllability via two kinds of controls is studied one kind of control is generated by boolean control networks another kind of control is free boolean sequences in both cases necessary and sufficient conditions of the controllability of boolean control networks are proved finally examples are given to illustrate the efficiency of the obtained results

on the periodic trajectories of boolean control networks abstract in this note we first characterize the periodic trajectories or equivalently the limit cycles of a boolean network and their global attractiveness we then investigate under which conditions all the trajectories of a boolean control network may be forced to converge to the same periodic trajectory if every trajectory can be driven to such a periodic trajectory this is possible by means of a feedback control law

system decomposition with respect to inputs for boolean control networks abstract in this paper the problem of system decomposition with respect to inputs is investigated for boolean control networks bcns first based on the linear representation of bcns some algebraic equivalent conditions for the decomposability are obtained second the concept of perfect equal vertex partition pevp is proposed for bcns a necessary and sufficient graphical condition for the decomposability i e the existence of a pevp is proposed third for calculating a pevp an algorithm called vertex set uniting algorithm vsua is designed finally a biological example is analyzed to show the effectiveness of the proposed method

the centre of generic algebras of small pi algebras abstract verbally prime algebras are important in pi theory they are well known over a field k of characteristic zero 0 and k t the trivial ones m n k m n e m a b e here k t is the free associative algebra with free generators t e is the infinite dimensional grassmann algebra over k m n k and m n e are the n n matrices over k and over e respectively moreover m a b e are certain subalgebras of m a b e defined below the generic algebras of these algebras have been studied extensively procesi gave a very tight description of the generic algebra of m n k the situation is rather unclear for the remaining nontrivial verbally prime algebras in this paper we study the centre of the generic algebra of m 11 e in two generators we prove that this centre is a direct sum of the field and a nilpotent ideal of the generic algebra we describe the centre of this algebra as a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by berele

least squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups let s a denote the orbit of a complex or real matrix a under a certain equivalence relation such as unitary similarity unitary equivalence unitary congruences etc efficient gradient flow algorithms are constructed to determine the best approximation of a given matrix a0 by the sum of matrices in s a1 s an in the sense of finding the euclidean least squares distance

pinning control design for the stabilization of boolean networks in this brief pinning control design for the stabilization of boolean network bn is addressed using the semitensor product of matrices transition matrix of the bn can be obtained we achieve global stability to the fixed point or the elementary cycle for the bn by changing the columns of the transition matrix then pinning nodes can be chosen and pinning control design algorithms are proposed finally an example is employed to show the proposed control design procedure

on the generation of symmetric lebesgue like points in the triangle we compute point sets on the triangle that have low lebesgue constant with sixfold symmetries and gauss legendre lobatto distribution on the sides up to interpolation degree 18 such points have the best lebesgue constants among the families of symmetric points used so far in the framework of triangular spectral elements

the spectrum of a hermitian matrix sum abstract an announcement by lidskii b v lidskii functional anal appl 10 1982 76 77 russian 139 140 english claimed to establish the explicit description of the spectrum of a hermitian matrix sum in the form conjectured by horn a horn pacific j math 16 1962 225 241 but no supporting proof has been published this paper begins an analysis of the claim and is the first step towards bridging the distance between the elementary methods of horn s 1962 paper and the partial solution of the same problem using noncommutative harmonic analysis by dooley et al i dooley r repka n wildberger lin mult alg 36 1993 79 102 the methods of horn lidskii lead to combinatorial issues of independent interest

gradient flow methods for matrix completion with prescribed eigenvalues matrix completion with prescribed eigenvalues is a special type of inverse eigenvalue problem the goal is to construct a matrix subject to both the structural constraint of prescribed entries and the spectral constraint of prescribed spectrum the challenge of such a completion problem lies in the intertwining of the cardinality and the location of the prescribed entries so that the inverse problem is solvable an intriguing question is whether matrices can have arbitrary entries at arbitrary locations with arbitrary eigenvalues and how to complete such a matrix constructive proofs exist to a certain point and those proofs such as the classical schur horn theorem are amazingly elegant enough in their own right beyond which very few theories or numerical algorithms are available in this paper the completion problem is recast as one of minimizing the distance between the isospectral matrices with the prescribed eigenvalues and the affined matrices with the prescribed entries the gradient flow is proposed as a numerical means to tackle the construction this approach is general enough that it can be used to explore the existence question when the prescribed entries are at arbitrary locations with arbitrary cardinalities

generalized frame operator distance problems abstract let s m d c be a positive semidefinite d d complex matrix and let a a i i i k r 0 k indexed by i k 1 k be a k tuple of positive numbers let t d a denote the set of families g g i i i k c d k such that g i 2 a i for i i k thus t d a is the product of spheres in c d endowed with the product metric for a strictly convex unitarily invariant norm n in m d c we consider the generalized frame operator distance function n s a defined on t d a given by n s a g n s s g where s g i i k g i g i m d c in this paper we determine the geometrical and spectral structure of local minimizers g 0 t d a of n s a in particular we show that local minimizers are global minimizers and that these families do not depend on the particular choice of n

state space analysis of boolean networks this paper provides a comprehensive framework for the state space approach to boolean networks first it surveys the authors recent work on the topic using semitensor product of matrices and the matrix expression of logic the logical dynamic equations of boolean control networks can be converted into standard discrete time dynamics to use the state space approach the state space and its subspaces of a boolean network have been carefully defined the basis of a subspace has been constructed particularly the regular subspace y friendly subspace and invariant subspace are precisely defined and the verifying algorithms are presented as an application the indistinct rolling gear structure of a boolean network is revealed

sampled data state feedback stabilization of boolean control networks in this letter we investigate the sampled data state feedback control sdsfc problem of boolean control networks bcns some necessary and sufficient conditions are obtained for the global stabilization of bcns by sdsfc different from conventional state feedback controls new phenomena observed the study of sdsfc based on the controllability matrix we derive some necessary and sufficient conditions under which the trajectories of bcns can be stabilized to a fixed point by piecewise constant control pcc it is proved that the global stabilization of bcns under sdsfc is equivalent to that by pcc moreover algorithms are given to construct the sampled data state feedback controllers numerical examples are given to illustrate the efficiency of the obtained results

brief paper controllability of boolean control networks via the perron frobenius theory boolean control networks bcns are recently attracting considerable interest as computational models for genetic and cellular networks addressing control theoretic problems in bcns may lead to a better understanding of the intrinsic control in biological systems as well as to developing suitable protocols for manipulating biological systems using exogenous inputs we introduce two definitions for controllability of a bcn and show that a necessary and sufficient condition for each form of controllability is that a certain nonnegative matrix is irreducible or primitive respectively our analysis is based on a result that may be of independent interest namely a simple algebraic formula for the number of different control sequences that steer a bcn between given initial and final states in a given number of time steps while avoiding a set of forbidden states

polynomial identities of algebras in positive characteristic the verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic p 2 little is known about them in previous papers we discussed some sharp differences between these two cases for the characteristic and we showed that the so called tensor product theorem is in part no longer valid in the second case in this paper we study the gelfand kirillov dimension of the relatively free algebras of verbally prime and related algebras we compute the gk dimensions of several algebras and thus obtain a new proof of the fact that the algebras m1 1 e and e e are not pi equivalent in characteristic p 2 furthermore we show that the following algebras are not pi equivalent in positive characteristic ma b e e and ma b e ma b e e and mc d e e when a b c d a b

inequalities and equalities for the cartesian decomposition of complex matrices let a and b be hermitian matrices and let c a ib inequalities and equalities for the eigenvalues and singular values of the matrices a b and c are discussed known results on inequalities are surveyed new results on equality cases are proved and open problems are mentioned

local lidskii s theorems for unitarily invariant norms abstract lidskii s additive inequalities both for eigenvalues and singular values can be interpreted as an explicit description of global minimizers of functions that are built on unitarily invariant norms with domains consisting of certain orbits of matrices under the action of the unitary group in this paper we show that lidskii s inequalities actually describe all global minimizers of such functions and that local minimizers are also global minimizers we use these results to obtain partial results related to local minimizers of generalized frame operator distances in the context of finite frame theory

observability reconstructibility and state observers of boolean control networks the aim of this paper is to introduce and characterize observability and reconstructibility properties for boolean networks and boolean control networks described according to the algebraic approach proposed by d cheng and co authors in the series of papers 3 6 7 and in the recent monography a complete characterization of these properties based both on the boolean matrices involved in the network description and on the corresponding digraphs is provided finally the problem of state observer design for reconstructible bns and bcns is addressed and two different solutions are proposed

controllability and observability of boolean control networks the controllability and observability of boolean control networks are investigated after a brief review on converting a logic dynamics to a discrete time linear dynamics with a transition matrix some formulas are obtained for retrieving network and its logical dynamic equations from this network transition matrix based on the discrete time dynamics the controllability via two kinds of inputs is revealed by providing the corresponding reachable sets precisely then the problem of observability is also solved by giving necessary and sufficient conditions

optimal control of boolean control networks in this paper we address the optimal control problem for boolean control networks bcns we first consider the problem of finding the input sequences that minimize a given cost function over a finite time horizon the problem solution is obtained by means of a recursive algorithm that represents the analogue for bcns of the difference riccati equation for linear systems we prove that a significant number of optimal control problems for bcns can be easily reframed into the present setup in particular the cost function can be adjusted so as to include penalties on the switchings provided that we augment the size of the bcn state variable in the second part of the paper we address the infinite horizon optimal control problem and we provide necessary and sufficient conditions for the problem solvability the solution is obtained as the limit of the solution over the finite horizon 0 t and it is always achieved in a finite number of steps finally the average cost problem over the infinite horizon investigated in optimal control of logical control networks y zhao ieee trans autom control vol 56 no 8 pp 1766 1776 aug 2011 is addressed by making use of the results obtained in the previous sections

computing fekete and lebesgue points simplex square disk we have computed point sets with maximal absolute value of the vandermonde determinant fekete points or minimal lebesgue constant lebesgue points on three basic bidimensional compact sets the simplex the square and the disk using routines of the matlab optimization toolbox we have obtained some of the best bivariate interpolation sets known so far

realization of boolean control networks based on the linear expression of the dynamics of boolean networks the coordinate transformation of boolean variables is defined it follows that the state space coordinate transformation for the dynamics of boolean networks is revealed using it the invariant subspace for a boolean control network is defined then the structure of a boolean control network is analyzed and the controllable and observable normal forms and the kalman decomposition form are presented finally the realization problem including minimum realization of boolean control networks is investigated

invariant theory for matrices over the grassmann algebra we prove that the algebra of polynomial maps mn e e invariant under simultaneous conjugation is generated by the queer trace polynomials

set stabilization for switched boolean control networks this paper studies the set stabilization of switched boolean control networks in which a feedback control design algorithm is presented by a constructive method necessary and sufficient conditions of the set stabilization of switched boolean control networks under arbitrary switching signals are presented for the cases of the control input relying on switching signals or not respectively furthermore the corresponding switching signal independent and switching signal dependent controllers are provided for these two cases respectively it is shown that the condition of the switching signal dependent controller is less conservative than the one of the switching signal independent controller finally examples are given to illustrate the effectiveness of the proposed results

on algorithms for state feedback stabilization of boolean control networks abstract this paper deals with the algorithms for state feedback stabilization of boolean control networks bcns by resorting to the semi tensor product stp technique the labelled digraph that can be used to completely characterize the dynamics of bcns is derived which leads to an equivalent graphical description for the stabilization of bcns what is more interesting is the fact that the existence of a state feedback control law stabilizing the bcn to some given equilibrium point can be characterized in terms of its spanning in tree consequently two in tree search algorithms namely the breadth first search and the depth first search are proposed to design the state feedback stabilizing law when global stabilization is feasible besides some basic properties about the tree search algorithms are addressed a biological example is employed to illustrate the applicability and usefulness of the developed algorithms

colength sequences for kemer algebras this paper is a sequel to 4 in that paper we calculated the asymptotic behavior of the colength sequence for k k matrices and in this paper we will calculate the asymptotic behavior of the colength sequence for the algebras mk given a p i algebra a the cocharacter sequence n a n 0 is a sequence of sncharacters if we decompose each character into a sum of irreducible sn characters we would get something of the form n a m a let ln a be the length of n a so ln a par n m a the sequence of numbers ln a is the colength sequence of a similar constructions can also be made for other cocharacter sequences such as the trace cocharacter if a is an algebra with trace kemer classified verbally prime p i algebras into three families see 8 one is k k matrices whose colength sequence we investigated in 4 one is k k matrices over an infinite dimensional grassmann algebra we do not know yet how to deal with this case and the third is the algebras mk let e be an infinite dimensional grassmann algebra and let k n then mk is a subalgebra of the matrix algebra mn e in order to define it note that the grassmann algebra has a natural z 2z grading e e0 e1 with respect to which it is anticommutative i e degree 0 elements are central and degree 1 elements anticommute with each other let i and j be between 1 and n we define the ordered pair i j to be of degree 0 if either 1 i j k or k 1 i j n and of degree 1 otherwise then mk will be the set of all matrices aij mn e in which the degree of each aij

colength sequences for matrices over grassmann algebras abstract we prove that the colength sequence of k k matrices over the grassmann algebra is asymptotic to a constant times n k 4

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

on the inertia of sums of hermitian matrices abstract given n n hermitian matrices h1 hp a complete description is found for the possible inertias of the sum h1 hp in terms of the inertias of hi i 1 p we also consider and solve other related problems

bisimulations of boolean control networks a framework for analyzing bisimulation relations of boolean control networks bcns is set up in this paper bisimulation relations are natural objectives in control systems theory a bisimulation relation between a pair of control systems defines a relation on their state spaces or state sets explaining how a trajectory or transition of one system can be paired with a trajectory or transition of the other system and vice versa the paper first formalizes the notion of bi simulation in bcns and presents detailed results characterizing bi simulation relations for bcns then as an application of the notion of bisimulation it considers the propagation of the fundamental properties of macro controllability and stabilizability through a bisimulation relation it thereby suggests the possibility that certain control properties of a bcn can be inferred by analyzing a potentially simpler bcn the analysis developed in this paper is based on the semitensor product approach which gives algorithms that have e

output tracking control of boolean control networks via state feedback this paper investigates the state feedback based output tracking control of boolean control networks bcns with a constant reference signal by using the semi tensor product method based on the algebraic expression of bcns and by constructing a series of reachable sets a general procedure is proposed for the design of the state feedback laws for bcns to track a constant reference signal the study of an illustrative example shows that the obtained new results are effective in designing state feedback based output tracking controllers for bcns to track a constant reference signal

stability robust stabilization and h control of singular impulsive systems via switching control abstract in this paper stability robust stabilization and h control of singular impulsive systems are studied some new fundamental properties are derived for switched singular systems subject to impulse effects applying the lyapunov function theory several sufficient conditions are established for exponential stability robust stabilization and h control of the corresponding singular impulsive closed loop systems some numerical examples are given to demonstrate the effectiveness of the proposed control and stabilization methods

boolean dynamics of networks with scale free topology the dynamics of boolean networks with scale free topology are studied the existence of a phase transition from ordered to chaotic dynamics governed by the value of the scale free exponent of the network is shown analytically by analyzing the overlap between two distinct trajectories the phase diagram shows that the phase transition occurs for values of the scale free exponent in the open interval 2 2 5 since the boolean networks under study are directed graphs the scale free topology of the input connections and that of the output connections are studied separately ultimately these two topologies are shown to be equivalent a numerical study of the attractor structure of the configuration space reveals that this structure is similar in both networks with scale free topologies and networks with homogeneous random topologies however an important result of this work is that the fine tuning usually required to achieve stability in the dynamics of networks with homogeneous random topologies is no longer necessary when the network topology is scale free finally based on the results presented in this work it is hypothesized that the scale free topology favors the evolution and adaptation of network functioning from a biological perspective

disturbance decoupling control design for switched boolean control networks abstract this paper investigates a kind of disturbance decoupling problems ddps for switched boolean control networks bcns that is the disturbance independent output decoupling problem via the semi tensor product of matrices by converting the dynamics of switched bcns into an algebraic form and using the redundant variable separation technique all possible state feedback controllers are designed for the disturbance independent output decoupling problem then based on the obtained state feedback disturbance decoupling controllers and the system s output equation a constructive procedure is proposed to design all possible output feedback disturbance decoupling controllers the study of two illustrative examples shows that the new results obtained in this paper are effective in designing disturbance decoupling controllers for switched bcns

further results on feedback stabilization control design of boolean control networks abstract this paper investigates designing all possible feedback stabilizers for boolean control networks bcns some new results on the feedback stabilization control design of bcns are presented the main tool used in the paper is the semi tensor product of matrices first the complete family of reachable sets is defined for bcns then it is shown that all the complete families of reachable sets determine all possible state feedback stabilizers second using all the complete families of reachable sets all possible state feedback stabilizers are obtained third a necessary and sufficient condition is obtained for the existence of output feedback stabilizers based on this condition all possible output feedback stabilizers are designed for bcns finally the obtained new results are applied to the regulation of the lactose operon in escherichia coli

graded identities for tensor products of matrix super algebras over the grassmann algebra abstract in this paper we study the graded identities satisfied by the superalgebras m a b over the grassmann algebra and by their tensor products these algebras play a crucial role in the theory developed by a kemer that led to the solution of the long standing specht problem it is well known that over a field of characteristic 0 the algebras m pr qs ps qr and m p q m r s satisfy the same ordinary polynomial identities by means of describing the corresponding graded identities we prove that the t ideal of the former algebra is contained in the t ideal of the latter furthermore the inclusion is proper at least in case r s 1 1 finally we deal with the graded identities satisfied by algebras of type m 2 n 1 2 n 1 and relate these graded identities to the ones of tensor powers of the grassmann algebra our proofs are combinatorial and rely on the relationship between graded and ordinary identities as well as on appropriate models for the corresponding relatively free graded algebras

stability and stabilization of boolean networks with impulsive effects abstract this paper investigates the stability and stabilization of boolean networks with impulsive effects after giving a survey on semi tensor product of matrices we convert a boolean network with impulsive effects into impulsive discrete time dynamics then some necessary and sufficient conditions are given for the stability and stabilization of boolean networks with impulsive effects finally examples are provided to illustrate the efficiency of the obtained results

function perturbation impact on feedback stabilization of boolean control networks function perturbation analysis of boolean networks is an important topic in the study of gene regulation due to gene mutation or immeasurable variables this brief studies the function perturbation impact on feedback stabilization of boolean control networks bcns by using the algebraic state space representation approach first the state feedback stabilization control design of bcns is recalled and the function perturbation problem is formulated second given a state feedback stabilizer it is robust to the considered function perturbation if one of the following three cases happens 1 the block where the function perturbation occurs is different from the block which is affected by the state feedback stabilizer case 1 2 when case 1 does not happen the perturbed column converges to the equilibrium faster than the original column case 2 3 when cases 1 and 2 do not happen the perturbed column does not belong to the reachable set of the original column case 3 third when the perturbed column belongs to the reachable set of the original column a constructive procedure is proposed to modify the given state feedback stabilizer to be robust to the function perturbation finally the obtained new results are applied to the function perturbation analysis of lactose operon in escherichia coli the main novelty of this brief is to develop a new theoretical framework for the robustness of feedback controllers of bcns with respect to function perturbation which is not solved in the existing literature

kalman decomposition for boolean control networks the kalman decomposition problem for boolean control networks bcns is solved completely a necessary and sufficient condition described by vertex partitions of digraphs is proposed for the kalman decomposition and an algorithm is designed to compute the logical coordinate transformation that results in the kalman decomposition finally two examples are presented one example illustrates the procedure of realizing the kalman decomposition the other one reveals that not all the bcns can be transformed into the kalman decomposition form this is an essential property different from that of the traditional linear control systems

the equivalence issue of two kinds of controllers in boolean control networks abstract in this paper the equivalence issue between state feedback controller and free sequence controller in boolean control network bcn is investigated based on the algebraic representation of boolean networks we prove that a boolean control network can be stabilized to a cycle or a fixed point via free sequence controller if and only if there is a state feedback controller which can derive the boolean control network to the cycle or the fixed point then two algorithms are provided to find the state feedback controller after that we prove that a bcn can be globally controllable by free sequence controller while the bcn is not necessarily globally controllable by state feedback controller at last an example is given to illustrate the results

polynomial fitting and interpolation on circular sections we construct weakly admissible polynomial meshes wams on circular sections such as symmetric and asymmetric circular sectors circular segments zones lenses and lunes the construction resorts to recent results on subperiodic trigonometric interpolation the paper is accompanied by a software package to perform polynomial fitting and interpolation at discrete extremal sets on such regions

polynomial approximation and cubature at approximate fekete and leja points of the cylinder the paper deals with polynomial interpolation least square approximation and cubature of functions dened on the rectangular cylinder k d 1 1 with d the unit disk the nodes used for these processes are the approximate fekete points afp and the discrete leja points dlp extracted from suitable weakly admissible meshes wams of the cylinder from the analysis of the growth of the lebesgue constants approximation and cubature errors we show that the afp and the dlp extracted from wam are good points for polynomial approximation and numerical integration of functions dened on the cylinder

global robust stability and stabilization of boolean network with disturbances abstract based on semi tensor product of matrices global robust stability and stabilization of boolean network bn with disturbances are investigated firstly the definition of global robust stability for bn with disturbances is presented by using the algebraic state space representation of disturbed bn some necessary and sufficient criteria are obtained to ensure the global robust stability with respect to w r t a fixed point or w r t a limit cycle in contrast if a given disturbed bn is not globally robust stable w r t a fixed point or w r t a limit cycle system can achieve stability by a matrix transformation technique however it is a difficult task to find such a suitable matrix transformation in this paper a pinning state feedback control design is proposed to find a suitable transformation the obtained results are well illustrated by numerical example

the detection and stabilisation of limit cycle for deterministic finite automata abstractin this paper the topological structure properties of deterministic finite automata dfa under the framework of the semi tensor product of matrices are investigated first the dynamics of dfa are converted into a new algebraic form as a discrete time linear system by means of boolean algebra using this algebraic description the approach of calculating the limit cycles of different lengths is given second we present two fundamental concepts namely domain of attraction of limit cycle and prereachability set based on the prereachability set an explicit solution of calculating domain of attraction of a limit cycle is completely characterised third we define the globally attractive limit cycle and then the necessary and sufficient condition for verifying whether all state trajectories of a dfa enter a given limit cycle in a finite number of transitions is given fourth the problem of whether a dfa can be stabilised to a limit cycle by the state feedback controller is discussed criteri

convex combinations of projections abstract on an n dimensional inner product space every operator t that satisfies o t i is a convex combination of as few as log 2 n 2 projections and this number is sharp if o t i and trace t is a rational number then t is an average of projections further results are also obtained for the cases when the projections are required to have the same rank and or to be commuting in each case the optimal number of projections is determined

controllability analysis and control design for switched boolean networks with state and input constraints this paper investigates the controllability analysis and the control design for switched boolean networks sbns with state and input constraints by using the semi tensor product method and presents a number of new results on their controllability optimal control and stabilization first the constrained incidence matrix is proposed for sbns based on which several necessary and sufficient conditions are obtained for the controllability second using the results on the controllability two algorithms are established to design proper switching sequence and controls that minimize the cost functional in a fixed designed shortest termination time third the constrained sbn is converted to an equivalent unconstrained one based on which some necessary and sufficient conditions are presented for the stabilization of the constrained sbn with open loop and closed loop controls respectively finally a practical example of the apoptosis network is studied by using the new results obtained in this paper the s

normalization and solvability of dynamic algebraic boolean networks in this brief we first study the normalization of dynamic algebraic boolean networks dabns a new expression for the normalized dabns is obtained as applications of this result the solvability and uniqueness of the solution to dabns are then investigated necessary and sufficient conditions for the solvability and the uniqueness are obtained in addition pinning control to ensure the solvability and uniqueness of the solution to dabns is also studied numerical examples are given to illustrate the efficiency of the proposed results

impulsive control of discrete systems with time delay this note deals with impulsive control of discrete systems with time delay it s the first time that an impulsive controller was introduced for discrete systems with time delay firstly some stability criteria for delay discrete systems with impulse effects are obtained by using lyapunov functionals and razumikhin technique these stability criteria show that if the impulses satisfy some conditions then the original stable systems can keep their stability property even with impulse effects then by using lyapunov functionals or lyapunov functions together with razumikhin technique two stability criteria for the discrete systems with time delay under impulsive control are given these two results show that unstable systems can be stable under impulsive control some examples are also presented to illustrate the efficiency of the obtained results

stabilization controllability and optimal control of boolean networks with impulsive effects and state constraints the present note investigates the stabilization controllability and optimal control problem of boolean networks with impulsive effects and state constraints by using the semi tensor product method the algebraic form of the boolean networks with impulsive effects and state constraints is derived the stabilization and controllability issues of the systems are investigated and some necessary and sufficient conditions are obtained in addition the mayer type optimal control problem is also studied and algorithms are provided to design the control sequence furthermore examples are given to illustrate the main results

function perturbations on singular boolean networks abstract this paper is devoted to studying function perturbations on the transition matrix and topological structure of a singular boolean network sbn via the semi tensor product of matrices first the algebraic form of an sbn is given and we discuss how the transition matrix of the sbn changes under function perturbations then the local uniqueness of solutions to the sbn is studied under which the impacts of function perturbations on the topological structure are investigated finally examples are given to show the effectiveness of the obtained results

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

graded polynomial identities of matrices we consider g graded polynomial identities of the p p matrix algebra mp k over a field k of characteristic 0 graded by an arbitrary group g we find relations between the g graded identities of the g graded algebra mp k and the g h graded identities of the tensor product of mp k and the h graded algebra mq k with a fine h grading we also find a basis of the g graded identities of mp k with an elementary grading such that the identity component coincides with the diagonal of mp k

set stability and set stabilization of boolean control networks based on invariant subsets this study addresses the set stability of boolean networks bns and set stabilization of boolean control networks bcns set stability determines whether a bn converges to a given subset whereas set stabilizability addresses the issue of whether a bcn can be stabilized to a given subset many problems can be viewed as special cases of set stability and set stabilization including synchronization partial stability and partial stabilization problems the concepts of invariant subset and control invariant subset are introduced then algorithms for the largest invariant subset and the largest control invariant subset contained in a given subset are proposed based on the invariant subsets obtained the necessary and sufficient conditions for set stability and set stabilizability are established and formulae are provided to calculate the shortest transient periods for respective initial states a design procedure is proposed for finding all the time optimal set stabilizers finally an example is used to show the application of the proposed results to the synchronization problem of bns

output feedback stabilization of boolean control networks in the paper output feedback control of boolean control networks bcns is investigated first necessary and sufficient conditions for the existence of a time invariant output feedback tiof law stabilizing the bcn to some equilibrium point are given and constructive algorithms to test the existence of such a feedback law are proposed two sufficient conditions for the existence of a stabilizing time varying output feedback tvof are then given finally an example concerning the lac operon in the bacterium escherichia coli is presented to illustrate the effectiveness of the proposed techniques