synchronization of arbitrarily switched boolean networks this paper investigates the complete synchronization problem for the drive response switched boolean networks sbns under arbitrary switching signals where the switching signals of the response sbn follow those generated by the drive sbn at each time instant first the definition of complete synchronization is introduced for the drive response sbns under arbitrary switching signals second the concept of switching reachable set starting from a given initial state set is put forward based on it a necessary and sufficient condition is derived for the complete synchronization of the drive response sbns last we give a simple algebraic expression for the switching reachable set in a given number of time steps and two computable algebraic criteria are obtained for the complete synchronization of the sbns a biological example is given to demonstrate the effectiveness of the obtained main results

universal finite size scaling functions for percolation on three dimensional lattices using a histogram monte carlo simulation method hmcsm hu lin and chen found that bond and site percolation models on planar lattices have universal finite size scaling functions for the existence probability e p the percolation probability p and the probability w n for the appearance of n percolating clusters in these models in this paper we extend above study to percolation on three dimensional lattices with various linear dimensions l using the hmcsm we calculate the existence probability e p and the percolation probability p for site and bond percolation on a simple cubic sc lattice and site percolation on body centered cubic and face centered cubic lattices each lattice has the same linear dimension in three dimensions using the data of e p and p in a percolation renormalization group method we find that the critical exponents obtained are quite consistent with the universality of critical exponents using a small number of nonuniversal metric factors we find that e p and p have universal finite size scaling functions this implies that the critical e p is a universal quantity which is 0 265 ifmmode pm else textpm fi 0 005 for free boundary conditions and 0 924 ifmmode pm else textpm fi 0 005 for periodic boundary conditions we also find that w n for site and bond percolation on sc lattices have universal finite size scaling functions

stability of the kauffman model random boolean networks the kauffman model are revisited by means of a novel decimation algorithm which removes variables that cannot be relevant to the asymptotic dynamics of the system the major part of the removed variables have the same fixed state in all limit cycles these variables are denoted as the stable core of the network and their number grows approximately linearly with n the number of variables in the original network the sensitivity of the attractors to perturbations is investigated we find that reduced networks lack the well known insensitivity observed in full kauffman networks we conclude that somewhat counterintuitive this remarkable property of full kauffman networks is generated by the dynamics of their stable core the decimation method is also used to simulate large critical kauffman networks for networks up to n 32 we perform full enumeration studies strong evidence is provided that the number of limit cycles grows linearly with n this result is in sharp contrast to the often cited square root of n behavior

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

observability of boolean control networks with state time delays this brief deals with the problem of the observability for the boolean control networks with time delays in states first using semi tensor product of matrices and the matrix expression of logic the boolean control networks with state delays can be converted into discrete time delay dynamics then the observability of the boolean control networks via two kinds of inputs is investigated by giving necessary and sufficient conditions finally examples are given to illustrate the efficiency of the obtained results

stability analysis for switched boolean networks under arbitrary switching signals this note investigates the stability of switched boolean networks sbns under arbitrary switching signal by using the tool of semi tensor product of matrices first an algebraic expression is given for the sbn via the semi tensor product based on it a necessary and sufficient condition is presented for the switching point reachability of the system second using the switching point reachability the pointwise stabilizability is studied and a necessary and sufficient condition is obtained finally the global stability of sbns under arbitrary switching signal is investigated and an easily verifiable necessary and sufficient condition is presented two illustrative examples are presented to show that the main results are effective in analyzing the stability of sbns

asymptotical stability of probabilistic boolean networks with state delays this paper devotes to establishing a bridge between asymptotical stability of a probabilistic boolean network pbn and a solution to its induced equations which are induced from the pbn s transition matrix by utilizing the semitensor product technique routinely the dynamics of a pbn with coincident state delays can be equivalently converted into that of a higher dimensional pbn without delays subsequently several novel stability criteria are derived from the standpoint of equations solution the most significant finding is that a pbn is globally asymptotically stable at a predesignated one point distribution if and only if a vector obtained by adding 1 at the bottom of this distribution is the unique nonnegative solution to pbn s induced equations moreover the influence of coincident state delays on pbn s asymptotical stability is explicitly analyzed without consideration of the convergence rate consequently such bounded state delays are verified to have no impact on pbn s stability albeit delays are time varying based on this worthwhile observation the time computational complexity of the aforementioned approach can be reduced by removing delays directly furthermore this universal procedure is summarized to reduce the time complexity of some previous results in the literature to a certain extent two examples are employed to demonstrate the feasibility and effectiveness of the obtained theoretical results

continuum percolation threshold for interpenetrating squares and cubes monte carlo simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions simulations are performed for two cases i objects whose edges are aligned parallel to one another and ii randomly oriented objects for squares whose edges are aligned the critical area fraction at the percolation threshold ensuremath varphi c 0 6666 ifmmode pm else textpm fi 0 0004 while for randomly oriented squares ensuremath varphi c 0 6254 ifmmode pm else textpm fi 0 0002 6 smaller for cubes whose edges are aligned the critical volume fraction at the percolation threshold ensuremath varphi c 0 2773 ifmmode pm else textpm fi 0 0002 while for randomly oriented cubes ensuremath varphi c 0 2168 ifmmode pm else textpm fi 0 0002 22 smaller

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

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

random walk percolation based modeling of two phase flow in porous media breakthrough time and net to gross ratio estimation abstract fluid flow modeling in porous media has many applications in waste treatment hydrology and petroleum engineering in any geological model flow behavior is controlled by multiple properties these properties must be known in advance of common flow simulations when uncertainties are present deterministic modeling often produces poor results percolation and random walk rw methods have recently been used in flow modeling their stochastic basis is useful in dealing with uncertainty problems they are also useful in finding the relationship between porous media descriptions and flow behavior this paper employs a simple methodology based on random walk and percolation techniques the method is applied to a well defined model reservoir in which the breakthrough time distributions are estimated the results of this method and the conventional simulation are then compared the effect of the net to gross ratio on the breakthrough time distribution is studied in terms of shannon entropy use of the entropy plot allows one to assign the appropriate net to gross ratio to any porous medium

identification of boolean control networks in this paper the identification of boolean control networks is addressed first necessary and sufficient conditions are obtained for the identification of state equation from input state data then a necessary and sufficient condition for a controllable boolean network to be observable is presented based on these two results a necessary and sufficient condition for the identification from input output data is achieved to practically identify the model a numerical algorithm is proposed two particular cases i identification of systems with a known network graph ii identification of a higher order boolean network are also investigated finally the approximate identification for large size networks is explored

asymptotic behavior of solutions for a three dimensional system of globally modified navier stokes equations with a locally lipschitz delay term abstract existence uniqueness and continuity properties of solutions for a globally modified version of navier stokes equations with finite delay terms within a locally lipschitz operator are established moreover we also analyze the stationary problem and under suitable assumptions we prove that there exists a unique stationary solution which is globally asymptotically exponentially stable

effect of anisotropy on finite size scaling in percolation theory we investigate the effects of anisotropy on finite size scaling of site percolation in two dimensions we consider a lattice of size n x ifmmode times else texttimes fi n y we define an aspect ratio ensuremath omega n x n y and consider the mean connected fraction p averaged over the realizations as a function of the site occupancy probability p the system size n x and this aspect ratio it is clear that there is an easy direction for percolation which is in the short direction i e y if ensuremath omega g1 and a difficult direction which is along the long axis we define an apparent percolation threshold in each direction as the value of p when 50 of realizations connect in that direction we show that standard finite size scaling applies if we use this apparent threshold we also find a finite size scaling for the fluctuations about this mean connected fraction

a maximum principle for single input boolean control networks boolean networks have recently been attracting considerable interest as computational models for genetic and cellular networks we consider a mayer type optimal control problem for a single input boolean network and derive a necessary condition for a control to be optimal this provides an analog of pontryagin s maximum principle for single input boolean networks

genetic networks with canalyzing boolean rules are always stable we determine stability and attractor properties of random boolean genetic network models with canalyzing rules for a variety of architectures for all power law exponential and flat in degree distributions we find that the networks are dynamically stable furthermore for architectures with few inputs per node the dynamics of the networks is close to critical in addition the fraction of genes that are active decreases with the number of inputs per node these results are based upon investigating ensembles of networks using analytical methods also for different in degree distributions the numbers of fixed points and cycles are calculated with results intuitively consistent with stability analysis fewer inputs per node implies more cycles and vice versa there are hints that genetic networks acquire broader degree distributions with evolution and hence our results indicate that for single cells the dynamics should become more stable with evolution however such an effect is very likely compensated for by multicellular dynamics because one expects less stability when interactions among cells are included we verify this by simulations of a simple model for interactions among cells

the kauffman model on small world topology abstract we apply kauffman s automata on small world networks to study the crossover between the short range and the infinite range case we perform accurate calculations on square lattices to obtain both critical exponents and fractal dimensions particularly we find an increase of the damage propagation and a decrease in the fractal dimensions when adding long range connections

brief paper controllability of probabilistic boolean control networks this paper deals with the controllability of probabilistic boolean control networks first a survey on the semi tensor product approach to probabilistic boolean networks is given second the controllability of probabilistic boolean control networks via two kinds inputs is studied finally examples are given to show the efficiency of the obtained results

solving fuzzy relational equations via semitensor product the problem of solving max min fuzzy relational equations is investigated first we show that if there is a solution then there is a corresponding solution within the set of parameters briefly the parameter set solution pss then the semitensor product of matrices is used to convert the logical equations into algebraic equations via the vector expression of logical variables under this form every pss can be obtained it is proved that all the solutions can be revealed from their corresponding pss some examples are presented to demonstrate the algorithm to solve fuzzy relational equations

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

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

policy iteration algorithm for optimal control of stochastic logical dynamical systems this brief investigates the infinite horizon optimal control problem for stochastic multivalued logical dynamical systems with discounted cost applying the equivalent descriptions of stochastic logical dynamics in term of markov decision process the discounted infinite horizon optimal control problem is presented in an algebraic form then employing the method of semitensor product of matrices and the increasing dimension technique a succinct algebraic form of the policy iteration algorithm is derived to solve the optimal control problem to show the effectiveness of the proposed policy iteration algorithm an optimization problem of p53 mdm2 gene network is investigated

the weak connectedness of the attainability set of weak solutions of the three dimensional navier stokes equations the attainability set of the weak solutions of the three dimensional navier stokes equations which satisfy an energy inequality is shown to be a weakly compact and weakly connected subset of the space h i e the kneser property holds in the weak topology for such weak solutions the proof of weak connectedness uses the strong connectedness of the attainability set of the weak solutions of the globally modified navier stokes equations which is first proved the weak connectedness of the weak global attractor of the three dimensional navier stokes equations is also established

driven stability of nonlinear feedback shift registers with inputs driven stable nonlinear feedback shift registers nfsrs with inputs are not only able to limit error propagations in convolutional decoders but also helpful to analyze the period properties of sequences generated by a cascade connection of nfsrs in stream ciphers an nfsr is driven stable if and only if the reachable set is a subset of the basin due to lack of efficient algebraic tools the driven stability of nfsrs with inputs has been much less studied this paper continues to address this research using a boolean control network approach viewing an nfsr with input as a boolean control network we first give its boolean control network representation which is characterized with a state transition matrix some properties of the state transition matrix are then provided based on these explicit forms are given for the reachable set and the set of basin two algorithms for obtaining both the sets are provided as well compared with the exhaustive search and the existing state operator method the boolean control network approach requires lower computational complexity for those nfsrs with their stages greater than 1

minimum time control of boolean networks boolean networks bns are discrete time dynamical systems with boolean state variables bns have recently been attracting considerable interest as computational models for biological systems and in particular gene regulating networks boolean control networks bcns are boolean networks with boolean inputs we consider the problem of steering a bcn from a given state to a desired state in minimal time using the algebraic state space representation assr of bcns we derive several necessary conditions stated in the form of maximum principles mps for a control to be time optimal in the assr every state and input vector is a canonical vector using this special structure yields an explicit state feedback formula for all time optimal controls to demonstrate the theoretical results we develop a bcn model for the genetic switch controlling the lambda phage development upon infection of a bacteria our results suggest that this biological switch is designed in a way that guarantees minimal time respon

controllability of probabilistic boolean control networks based on transition probability matrices in this paper we propose a new approach to investigate the controllability and reachability of probabilistic boolean control networks pbcns with forbidden states we first give a simple algebraic formula for the transition probability between two states in a given number of time step while avoiding a set of forbidden states then we construct the controllability matrix based on a new operator and some necessary and sufficient conditions are obtained for the controllability and reachability of pbcns a numerical example is given to illustrate the efficiency of the obtained results

optimal control of logical control networks this paper considers the infinite horizon optimal control of logical control networks including boolean control networks as a special case using the framework of game theory the optimal control problem is formulated in the sight of the algebraic form of a logical control network its cycles can be calculated algebraically then the optimal control is revealed over a certain cycle when the games using memory 1 which means the players only consider previous steps action at each step are considered the higher order logical control network is introduced and its algebraic form is also presented which corresponds to a conventional logical control network i e 1 then it is proved that the optimization technique developed for conventional logical control networks is also applicable to this memory case

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

navier stokes equations with delays on unbounded domains abstract some results on the existence and uniqueness of solutions to navier stokes equations when the domain is unbounded and the external force contains some hereditary characteristics are proved for both the evolutionary and the stationary cases exponential stability of the stationary solution is also established in dimension two

a matrix approach to graph maximum stable set and coloring problems with application to multi agent systems using the semi tensor product of matrices this paper investigates the maximum weight stable set and vertex coloring problems of graphs with application to the group consensus of multi agent systems and presents a number of new results and algorithms firstly by defining a characteristic logical vector and using the matrix expression of logical functions an algebraic description is obtained for the internally stable set problem based on which a new algorithm to find all the internally stable sets is established for any graph secondly the maximum weight stable set problem is considered and a necessary and sufficient condition is presented by which an algorithm to find all the maximum weight stable sets is obtained thirdly the vertex coloring problem is studied by using the semi tensor product method and two necessary and sufficient conditions are proposed for the colorability based on which a new algorithm to find all the k coloring schemes and minimum coloring partitions is put forward finally the obtained results are applied to multi agent systems and a new protocol design procedure is presented for the group consensus problem the study of illustrative examples shows that the results algorithms presented in this paper are very effective

complete synchronization of boolean networks we examine complete synchronization of two deterministic boolean networks bns coupled unidirectionally in the drive response configuration a necessary and sufficient criterion is presented in terms of algebraic representations of bns as a consequence we show that complete synchronization can occur only between two conditionally identical bns when the transition matrix of the drive network is nonsingular two examples are worked out to illustrate the obtained results

symbolic dynamics of boolean control networks we consider boolean control networks bcns and in particular boolean networks bns in the framework of symbolic dynamics sd we show that the set of state space trajectories of a bcn is a shift space of finite type sft this observation allows us to extend two important analysis tools from sd namely the artin mazur zeta function and the topological entropy to bns and bcns some of the theoretical results are illustrated using a bcn model of the core network regulating the mammalian cell cycle

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

synchronization for the realization dependent probabilistic boolean networks this paper investigates the synchronization problem for the realization dependent probabilistic boolean networks pbns coupled unidirectionally in the drive response configuration the realization of the response pbn is assumed to be uniquely determined by the realization signal generated by the drive pbn at each discrete time instant first the drive response pbns are expressed in their algebraic forms based on the semitensor product method and then a necessary and sufficient condition is presented for the synchronization of the pbns second by resorting to a newly defined matrix operator the reachable set from any initial state is expressed by a column vector consequently an easily computable algebraic criterion is derived assuring the synchronization of the drive response pbns finally three illustrative examples are employed to demonstrate the applicability and usefulness of the developed theoretical results

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

asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3d globally modified navier stokes equations abstract we first prove the existence and regularity of the trajectory attractor for a three dimensional system of globally modified navier stokes equations then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space in our construction the trajectory statistical solution is an invariant borel probability measure which is supported by the trajectory attractor and is invariant under the action of the translation semigroup as a byproduct of the regularity of the trajectory attractor we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions

scaling and universality in the spanning probability for percolation we discuss the spanning percolation probability function for three different spanning rules in general dimensions with both free and periodic boundary conditions our discussion relies on the renormalization group theory which indicates that apart from a few scale factors the scaling functions for the spanning probability are determined by the fixed point and therefore are universal for every system with the same dimensionality spanning rule aspect ratio and boundary conditions for square and rectangular systems we combine this theory with simple relations between the spanning rules and with duality arguments and find strong relations among different derivatives of the spanning function with respect to the scaling variables thus yielding several universal amplitude ratios and allowing a systematic study of the corrections to scaling both singular and analytic in the system size the theoretical predictions are numerically confirmed with excellent accuracy textcopyright 1996 the american physical society

stability results for 2d navier stokes equations with unbounded delay abstract some results related to 2d navier stokes equations when the external force contains hereditary characteristics involving unbounded delays are analyzed first the existence and uniqueness of solutions is proved by galerkin approximations and the energy method the existence of stationary solution is then established by means of the lax milgram theorem and the schauder fixed point theorem the local stability analysis of stationary solutions is studied by several different methods the classical lyapunov function method the razumikhin lyapunov technique and by constructing appropriate lyapunov functionals finally we also verify the polynomial stability of the stationary solution in a particular case of unbounded variable delay exponential stability in this infinite delay setting remains as an open problem

evolution by damage spreading in kauffman model in kauffman s boolean automata model on the square lattice the darwinian fitness of survival can be defined as the fraction of elements which do not change from one iteration to the next biological mutations are simulated by filpping one bit in the rule of one site selection of the fitter mutant then optimizes the whole lattice completely this optimization is particularly effective near the critical point of the transition to chaos but is in itself not a critical phenomenon also a two dimensional spin glass can be optimized in this way

random boolean network model exhibiting deterministic chaos this paper considers a simple boolean network with n nodes each node s state at time t being determined by a certain number of parent nodes which may vary from one node to another this is an extension of a model studied by andrecut and ali int j mod phys b 15 17 2001 who consider the same number of parents for all nodes we make use of the same boolean rule as andrecut and ali provide a generalization of the formula for the probability of finding a node in state 1 at a time t and use simulation methods to generate consecutive states of the network for both the real system and the model the results match well we study the dynamics of the model through sensitivity of the orbits to initial values bifurcation diagrams and fixed point analysis we show that the route to chaos is due to a cascade of period doubling bifurcations which turn into reversed period halving bifurcations for certain combinations of parameter values

finite size scaling for percolation on elongated lattices in two and three dimensions we derive scaling laws for the percolation properties of an elongated lattice i e those with dimensions of l d ensuremath 1 ifmmode times else texttimes fi mathrm nl in d dimensions where n denotes the aspect ratio of the lattice based on statistical arguments it is shown that in the direction of the extension the percolation threshold scales approximately as mathrm ln n 1 a in both two and three dimensions extensive monte carlo simulations of the site percolation model confirm this scaling behavior it is further shown that the density of the incipient infinite cluster at the percolation threshold scales differently in two and three dimensions

model construction of boolean network via observed data in this paper a set of data is assumed to be obtained from an experiment that satisfies a boolean dynamic process for instance the dataset can be obtained from the diagnosis of describing the diffusion process of cancer cells with the observed datasets several methods to construct the dynamic models for such boolean networks are proposed instead of building the logical dynamics of a boolean network directly its algebraic form is constructed first and then is converted back to the logical form firstly a general construction technique is proposed to reduce the size of required data the model with the known network graph is considered motivated by this the least in degree model is constructed that can reduce the size of required data set tremendously next the uniform network is investigated the number of required data points for identification of such networks is independent of the size of the network finally some principles are proposed for dealing with data with errors

on reachability and controllability of switched boolean control networks this paper investigates the reachability and controllability of switched boolean control networks bcns by using the semi tensor product method first a kind of switching input state transfer graph is defined based on which and the algebraic expression of switched bcns we thus propose a new kind of switching input state incidence matrix that contains the complete information of the switching input state transfer graph second by the switching input state incidence matrix several necessary and sufficient conditions are obtained for the reachability and controllability of switched bcns finally an algorithm is established to find a switching sequence and a control scheme to realize the reachability of switched bcns in the shortest time the study of an illustrative example shows that the new results obtained in this paper are very effective in investigating the reachability and controllability of switched bcns

phase transitions in random networks simple analytic determination of critical points the standard method for determining the critical points in random networks such as boolean nets is based on the derrida approximation this method leads to the critical points based on the so called annealed approximation in this paper we present a very simple annealed approximation based on the study of damage spreading when single elements are modified several examples are analyzed

on boolean control networks with maximal topological entropy boolean control networks bcns are discrete time dynamical systems with boolean state variables and inputs that are interconnected via boolean functions bcns are recently attracting considerable interest as computational models for genetic and cellular networks with exogenous inputs the topological entropy of a bcn with m inputs is a nonnegative real number in the interval 0 m log 2 roughly speaking a larger topological entropy means that asymptotically the control is more powerful we derive a necessary and sufficient condition for a bcn to have the maximal possible topological entropy our condition is stated in the framework of cheng s algebraic state space representation of bcns this means that verifying this condition incurs an exponential time complexity we also show that the problem of determining whether a bcn with n state variables and m n inputs has a maximum topological entropy is np hard suggesting that this problem cannot be solved in general using a polynomial time algorithm

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

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory we investigate the effects of anisotropy on the finite size scaling of connectivity and conductivity of continuum percolation in three dimensions we consider a system of size x ifmmode times else texttimes fi y ifmmode times else texttimes fi z in which cubic bodies of size a ifmmode times else texttimes fi b ifmmode times else texttimes fi c are placed randomly we define two aspect ratios to request anisotropy then we expect that the displacement of average connected fraction p averaged over the realizations about the isotropic universal curves will be a function of the two aspect ratios this is accounted by considering an apparent percolation threshold in each direction which leads to 50 of realizations connecting in that direction we find the aspect ratios dependency of the apparent threshold and investigate the finite size scaling transformations for the mean connected fraction and its associated fluctuations moreover we apply a single phase pressure solver to determine the conductivity of various realizations of the system finally we apply the same idea to account for the effect of anisotropy on the conductivity scaling

exact solution of kauffman s model with connectivity one kauffman s model is a randomly assembled network of boolean automata each automaton receives inputs from at most k other automata its state at discrete time t 1 is determined by a randomly chosen but fixed boolean function of the k inputs at time t the resulting quenched random dynamics of the network demonstrates two phases a frozen and a chaotic phase the authors give an exact solution of the model for connectivity k 1 valid everywhere in the frozen phase and at a critical point valid for finite as well as for infinite networks they discuss the network s critical behaviour and finite size effects the results for the frozen phase presented complement recent exact results for the chaotic phase obtained for k infinity

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

state feedback stabilization for probabilistic boolean networks abstract a probabilistic boolean network pbn is a discrete time system composed of a family of boolean networks bns between which the pbn switches in a stochastic fashion studying control related problems in pbns may provide new insights into the intrinsic control in biological systems and enable us to develop strategies for manipulating complex biological systems using exogenous inputs this paper investigates the problem of state feedback stabilization for pbns based on the algebraic representation of logic functions a necessary and sufficient condition is derived for the existence of a globally stabilizing state feedback controller and a control design method is proposed when the presented condition holds it is shown that the controller designed via the proposed procedure can simultaneously stabilize a collection of pbns that are composed of the same constituent bns

disturbance decoupling of boolean control networks disturbance decoupling problem ddp of boolean control networks is considered using semi tensor product of matrices and the matrix expression of logical functions a working procedure is proposed to solve the problem this procedure consists of two key design steps first how to convert a system into an output friendly coordinate frame an algorithm is provided to calculate the output friendly subspaces secondly it was shown how to find proper controllers to solve the problem if it is solvable a state variable separation form is introduced to guide the design of controllers based on the design technique necessary and sufficient conditions are obtained for the solvability of ddp

the strange man in random networks of automata abstract we have performed computer simulations of kauffman s automata on several graphs such as the regular square lattice and invasion percolation clusters in order to investigate phase transitions radial distributions of the mean total damage dynamical exponent and propagation speeds of the damage when one adds a damaging agent nicknamed strange man despite the increase in the damaging efficiency we have not observed any appreciable change of the transition threshold to chaos neither for the short range nor for the small world case on the square lattices when the strange man is added in comparison to when small initial damages are inserted in the system particularly we have checked the damage spreading when some connections are removed on the square lattice and when one considers special invasion percolation clusters high boundary saturation clusters it is seen that the propagation speed in these systems is quite sensible to the degree of dilution on the square lattice and to the degree of saturation on invasion percolation clusters

effects of alternative connectivity on behavior of randomly constructed boolean networks random boolean networks rbns were first introduced and used to model regulatory gene networks by kauffman in 1969 and since then have been frequently used to model networks at various levels of biological organization the standard model has a characteristic topology in which the network has n genes with an average of k input connections selected at random we have used rbns to examine regulatory gene networks with four different topologies which are characterized by different rank distributions of output connections that vary from uniform to highly skewed among these is the topology of the standard model which is included for purposes of comparison and a topology with a power law rank distribution which is based on recent data for the regulatory gene network of the bacterium escherichia coli we also examine effects of bias in the distribution of boolean functions for the network the dynamical properties and mutual information of these networks depend not only on their size but also on their topology and boolean functions networks with the more uniform rank distributions exhibit longer lengths of attractors larger numbers of attractors and less mutual information networks with the more skewed rank distributions have complementary properties when viewed as biological decision making networks those with either the most uniform or the most skewed rank distributions have disadvantageous properties the intermediate rank distribution exhibited by the regulatory gene network of e coli avoids these disadvantages

state feedback based output tracking control of probabilistic boolean networks abstract this paper studies the state feedback based output tracking control of probabilistic boolean control networks pbcns by using the semi tensor product of matrices a series of reachable sets with probability 1 is defined inductively for pbcns and some useful properties are obtained for the reachable sets based on the properties of the reachable sets a necessary and sufficient condition is presented for the state feedback based output tracking control of pbcns meanwhile a constructive procedure is proposed to design state feedback laws for pbcns to track a constant reference signal the study of two illustrative examples shows that the obtained new results are very effective