scalar equations for synchronous boolean networks with biological applications one way of coping with the complexity of biological systems is to use the simplest possible models which are able to reproduce at least some nontrivial features of reality although two value boolean models have a long history in technology it is perhaps a little bit surprising that they can also represent important features of living organizms in this paper the scalar equation approach to boolean network models is further developed and then applied to two interesting biological models in particular a linear reduced scalar equation is derived from a more rudimentary nonlinear scalar equation this simpler but higher order two term equation gives immediate information about both cycle and transient structure of the network

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

finite time stability analysis of stochastic switched boolean networks with impulsive effect abstract this paper analyzes the finite time stability of stochastic switched boolean networks sbns with impulsive effect using the algebraic state space representation assr approach the dynamics of sbns with impulsive effect is converted to an algebraic form based on the algebraic form some necessary and sufficient conditions are presented for the finite time stability of stochastic sbns with impulsive effect including probabilistic switching signal case and markov jump switching signal case the study of an illustrative example shows that the obtained new results are effective

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

invariant subset and set stability of boolean networks under arbitrary switching signals this study investigates the set stability of switched boolean networks bns with arbitrary switching signals based on invariant subsets set stability determines whether a bn converges to a given subset while arbitrary switching signals can be used to characterize uncertainties and disturbances an algorithm is proposed to calculate the largest invariant subset contained in any given set as this is the key step in establishing the set stability criterion based on this algorithm a necessary and sufficient condition for set stability is obtained the result is then used to solve the robust output synchronization problem and the robust node synchronization problem for bns in the presence of uncertainties and disturbances examples are presented to demonstrate the use of the proposed approach

quantities characterizing semi fredholm operators and perturbation radii abstract we show that none of the operational quantities that have been introduced to characterize the upper or the lower semi fredholm operators is equivalent to the corresponding perturbation radius

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

an effective z stretching method for paraxial light beam propagation simulations a z stretching finite difference method is developed for simulating the paraxial light beam propagation through a lens in a cylindrically symmetric domain by introducing a domain transformation in the z direction we solve the corresponding complex difference equations containing an interface singularity over a computational space for great simplicity and efficiency a specially designed matrix analysis is constructed to the study the numerical stability computational experiments are carried out for demonstrating our 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

state analysis of boolean control networks with impulsive and uncertain disturbances in this paper we put forward a method for the state analysis of boolean control networks with impulsive and uncertain disturbances in detail we introduce the probability in the analysis of the state of boolean networks mathematically the stp as the analytical tool has be used for the modeling of uncertainty in addition we give a mathematical proof for the model lastly we conduct some simulations which validate the efficiency for this model

essentially compact schemes for unsteady viscous incompressible flows a new fourth order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced the scheme is based on the vorticity stream function formulation it is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions it is also very efficient at every time step or runge kutta stage only two poisson like equations have to be solved the poisson like equations are amenable to standard fast poisson solvers usually designed for second order schemes detailed comparison with the second order scheme shows the clear superiority of this new fourth order scheme in resolving both the boundary layers and the gross features of the flow this efficient fourth order scheme also made it possible to compute the driven cavity flow at reynolds number 106on a 10242grid at a reasonable cost fourth order convergence is proved under mild regularity requirements this is the first such result to our knowledge

transparent boundary conditions for time dependent problems a new approach to derive transparent boundary conditions tbcs for dispersive wave schrodinger heat and drift diffusion equations is presented it relies on the pole condition and distinguishes between physically reasonable and unreasonable solutions by the location of the singularities of the laplace transform of the exterior solution here the laplace transform is taken with respect to a generalized radial variable to obtain a numerical algorithm a mobius transform is applied to map the laplace transform onto the unit disc in the transformed coordinate the solution is expanded into a power series finally equations for the coefficients of the power series are derived these are coupled to the equation in the interior and yield transparent boundary conditions numerical results are presented in the last section showing that the error introduced by the new approximate tbcs decays exponentially in the number of coefficients

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

simultaneous stabilization for a set of boolean control networks abstract this paper investigates simultaneous stabilization of a collection of boolean control networks bcns by using the semi tensor product method and presents a number of new results first an algebraic expression of the bcns is obtained by the semi tensor product based on which some necessary and sufficient conditions are presented to solve the simultaneous stabilization problem by a free control sequence a state feedback control and an output feedback control respectively second using the column stacking form of matrices a new procedure is established to design both state feedback and output feedback controllers for the simultaneous stabilization problem the study of two illustrative examples shows that the new results obtained in this paper are very effective in solving simultaneous stabilization of a collection of bcns

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

numerical stabilities study of a decomposed compact method for highly oscillatory helmholtz equations abstract this paper concerns numerical stabilities of a decomposed compact finite difference method for solving helmholtz partial differential equation problems with large wave numbers radially symmetric transverse fields and standard polar coordinates are considered a decomposition is implemented to remove the anticipated singularity in the transverse direction compact scheme structures are introduced in the transverse direction to raise the accuracy for laser and subwavelength optical computations it is proven that while the highly accurate compact algorithm shies away from the conventional stability in the von neumann sense it is asymptotically stable with index one additionally a discussion on theoretical and numerical stabilities for this method is presented numerical experiments further demonstrate the high reliability of the scheme when implemented in highly oscillatory optical self focusing wave propagation simulations

an exponential transformation based splitting method for fast computations of highly oscillatory solutions splitting or decomposition methods have been widely used for achieving higher computational efficiency in solving wave equations a major concern has remained however if the wave number involved is exceptionally large in the case merits of a conventional splitting method may diminish due to the fact that tiny discretization steps need to be employed to compensate high oscillations this paper studies an alternative way for solving highly oscillatory paraxial wave problems via a modified splitting strategy in the process an exponential transformation is first introduced to convert the underlying differential equation to coupled nonlinear equations then the resulted oscillation free system is treated by a local one dimensional lod scheme for desired accuracy efficiency and computability the splitting method acquired is asymptotically stable and easy to use computational experiments are given to illustrate our numerical procedures

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

input state incidence matrix of boolean control networks and its applications the input state incidence matrix of a control boolean network is proposed it is shown that this matrix contains complete information of the input state mapping using it an easily verifiable necessary and sufficient condition for the controllability of a boolean control network is obtained the corresponding control which drives a point to a given reachable point is designed moreover certain topological properties such as the fixed points and cycles of a boolean control network are investigated then as another application a sufficient condition for the observability is presented finally the results are extended to mix valued logical control systems

a continuing exploration of a decomposed compact method for highly oscillatory wave problems this paper concerns a highly effective and decomposed compact scheme for solving a highly oscillatory paraxial helmholtz problem in radially symmetric fields the decomposition is utilized in the transverse direction to eliminate the singularity of the differential equation in polar coordinates numerical stability of the splitting scheme is investigated it is shown that the numerical method introduced is not only highly accurate and efficient due to its straightforward algorithmic structure but also stable under reasonable constraints for practical applications computational examples are presented to illustrate our conclusions

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

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

a domain decomposition method for the helmholtz equation in a multilayer domain the two dimensional helmholtz equation for problems where the physical domain consists of layers with different material properties is studied an efficient preconditioner for iterative solution of the problem is constructed the problem is discretized with fourth order accurate finite difference operators for the construction of the radiation boundary conditions a fourth order finite element method also is used the large sparse complex indefinite and ill conditioned system of equations that arises is solved with preconditioned restarted gmres a domain decomposition method is used in which the preconditioning is based on the schur complement algorithm with fast poisson type preconditioners for the subdomains the memory requirements for the preconditioner are nearly linear in the number of unknowns the arithmetic complexity for each iteration is low whereas the construction of the preconditioner is a bit more expensive electromagnetic wave propagation in a three layered waveguide is used as a model problem numerical experiments show that convergence is achieved in a few iterations compared with banded gaussian elimination which is a standard solution method the iterative method shows significant gain in both memory requirements and arithmetic complexity furthermore the relative gain grows when the problem size increases

stability and set stability in distribution of probabilistic boolean networks we propose a new concept stability in distribution sd of a probabilistic boolean network pbn which determines whether the probability distribution converges to the distribution of the target state namely a one point distributed random variable in a pbn stability with probability one stability in the stochastic sense and sd are equivalent the sd is easily generalized to subset stability i e to set stability in distribution ssd we prove that the transition probability from any state to an invariant subset or to a fixed point is nondecreasing in time this monotonicity is an important property in establishing stability criteria and in calculating or estimating the transient period we also obtain a verifiable necessary and sufficient condition for sd of pbns with independently and identically distributed switching we then show that sd problems of pbns with markovian switching and pbn synchronizations can be recast as ssd problems of markov chains after calculating the largest invariant subset of a markov chain in a given set by the newly proposed algorithm we propose a necessary and sufficient condition for ssds of markov chains the proposed method and results are supported by examples

global stability and stabilization of switched boolean network with impulsive effects the present paper investigates the global stability and stabilization of switched boolean network with impulsive effects by using the semi tensor product the algebraic form of the switched boolean network with impulsive effects is obtained then sufficient and necessary condition is presented for the global stability of the switched boolean network with impulsive effects furthermore sufficient and necessary conditions are provided for the stabilization of the switched boolean network with impulsive effects under two types of controls finally examples are presented to illustrate the main results

a linear representation of dynamics of boolean networks a new matrix product called semi tensor product of matrices is reviewed using it a matrix expression of logic is proposed where a logical variable is expressed as a vector a logical function is expressed as a multiple linear mapping under this framework a boolean network equation is converted into an equivalent algebraic form as a conventional discrete time linear system analyzing the transition matrix of the linear system formulas are obtained to show a the number of fixed points b the numbers of cycles of different lengths c transient period for all points to enter the set of attractors and d basin of each attractor the corresponding algorithms are developed and used to some examples

output controllability and optimal output control of state dependent switched boolean control networks in the present paper we investigate the output controllability and optimal output control problems of a state dependent switched boolean control network by using the semi tensor product the algebraic form of the system is obtained then output controllability problems of the system are discussed and some necessary and sufficient conditions are given next the mayer type optimal output control issue is considered and an algorithm is provided to find out the control sequence at last an example is given to show the effectiveness of the main 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

state feedback controller design for the synchronization of boolean networks with time delays abstract state feedback control design to make the response boolean network synchronize with the drive boolean network is far from being solved in the literature motivated by this this paper studies the feedback control design for the complete synchronization of two coupled boolean networks with time delays a necessary condition for the existence of a state feedback controller is derived first then the feedback control design procedure for the complete synchronization of two coupled boolean networks is provided based on the necessary condition finally an example is given to illustrate the proposed design procedure

control of large scale boolean networks via network aggregation a major challenge to solve problems in control of boolean networks is that the computational cost increases exponentially when the number of nodes in the network increases we consider the problem of controllability and stabilizability of boolean control networks address the increasing cost problem by partitioning the network graph into several subnetworks and analyze the subnetworks separately easily verifiable necessary conditions for controllability and stabilizability are proposed for a general aggregation structure for acyclic aggregation we develop a sufficient condition for stabilizability it dramatically reduces the computational complexity if the number of nodes in each block of the acyclic aggregation is small enough compared with the number of nodes in the entire boolean network