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

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

the steklov eigenvalue problem in a cuspidal domain in this paper we analyze the approximation by piecewise linear finite elements of a steklov eigenvalue problem in a plane domain with an external cusp this problem is not covered by the literature and its analysis requires a special treatment indeed we develop new trace theorems and we also obtain regularity results for the source counterpart moreover under appropriate assumptions on the meshes we present interpolation error estimates for functions in fractional sobolev spaces these estimates allow us to obtain appropriate convergence results of the source counterpart which in the context of the theory of compact operator are a fundamental tool in order to prove the convergence of the eigenpairs at the end we prove the convergence of the eigenpairs by using graded meshes and present some numerical tests

synchronization in an array of output coupled boolean networks with time delay this brief presents an analytical study of synchronization in an array of coupled deterministic boolean networks bns with time delay two kinds of models are considered in one model the outputs contain time delay while in another one the outputs do not one restriction in this brief is that the state delay and output delay are restricted to be equal by referring to the algebraic representations of logical dynamics and using the techniques of semitensor product of matrices some necessary and sufficient conditions are derived for the synchronization of delay coupled bns examples including a practical epigenetic example are given for illustration

random recursive construction of self similar fractal measures the noncompact case the self similarity of sets measures is often defined in a constructive way in the present paper it will be shown that the random recursive construction model of falconer graf and mauldin williams for statistically self similar sets may be generalized to the noncompact case we define a sequence of random finite measures which converges almost surely to a self similar random limit measure under certain conditions on the generating lipschitz maps we determine the carrying dimension of the limit measure

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

statistically self similar fractals letx be a complete separable bounded metric space and a borel probability measure on the space con x n of alln tuples of contractions ofx with the topology of pointwise convergence then there exists a unique self similar probability measurep on the spacek x of all non empty compact subsets ofx here a measurep onk x is called self similar if for every borel setb k x p b int p n left k o k n 1 left bigcup limits i 0 n 1 s i k i in b right right d mu s o s n 1 if for a e s 0 sn 1 eachs i has an inverse which satisfies a lipschitz condition then there is an 0 such that forp a e k k x the hausdorff dimensionh dim k is equal to ifx d is compact and has non empty interior and if a e s 0 sn 1 consists of similarities which satisfy a certain disjointness condition w r t x then is determined by the equation int sum limits i 0 n 1 lip s i alpha d mu s o s n 1 1 where lip s i denotes the smallest lipschitz constant fors i under fairly general assumptions the dimensional hausdorff measure ofp a e k k x equals o if andx are chosen in a rather special way thenp a e k k x is the graph of a homeomorphism of 0 1 or a curve or the graph of a continuous function

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

dimensions and singular traces for spectral triples with applications to fractals given a spectral triple a h d the functionals on a of the form a a d are studied where is a singular trace and is a generalised limit when is the dixmier trace the unique exponent d giving rise possibly to a non trivial functional is called hausdorff dimension and the corresponding functional the d dimensional hausdorff functional it is shown that the hausdorff dimension d coincides with the abscissa of convergence of the zeta function of d 1 and that the set of s for which there exists a singular trace giving rise to a non trivial functional is an interval containing d moreover the endpoints of such traceability interval have a dimensional interpretation the functionals corresponding to points in the traceability interval are called hausdorff besicovitch functionals these definitions are tested on fractals in r by computing the mentioned quantities and showing in many cases their correspondence with classical objects in particular for self similar fractals the traceability interval consists only of the hausdorff dimension and the corresponding hausdorff besicovitch functional gives rise to the hausdorff measure more generally for any limit fractal the described functionals do not depend on the generalized limit

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

state feedback stabilization for boolean control networks state feedback stabilization for boolean control networks is investigated in this technical note based on the algebraic representation of logical dynamics in terms of the semi tensor product of matrices a necessary and sufficient condition is derived for the existence of a globally stabilizing state feedback controller and a general control design approach is proposed when global stabilization is feasible via state feedback instead of designing the logical form of a stabilizing feedback law directly we first construct its algebraic representation and then convert the algebraic representation back to the logical form an example is worked out to illustrate the proposed design procedure

tangent measure distributions of fractal measures tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in euclidean spaces the idea is loosely speaking to attach to every point of the set a family of random measures called the alpha dimensional tangent measure distributions at the point which describe asymptotically the alpha dimensional scenery seen by an observer zooming down towards this point this tool has been used by bandt ba and graf g to study the regularity of the local geometry of self similar sets but in this paper we show that its scope goes much beyond this situation and in fact it may be used to describe a strong regularity property possessed by every measure we show that for every measure mu on a euclidean space and any dimension alpha at mu almost every point all alpha dimensional tangent measure distributions are palm measures this means that the local geometry of every dimension of general measures can be described like the local geometry of self similar sets by means of a family of statistically self similar random measures we believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities

self similar random measures a set is called self similar if it is decomposable into parts which are similar to the whole this notion was generalized to random sets in the present paper an alternative axiomatic approach is given which makes precise the following idea using palm distribution theory a random set is statistically self similar if it is statistically scale invariant with respect to any center chosen at random from that set for these sets hausdorff dimension coincides with an intrinsic self similarity index

observability of boolean networks a graph theoretic approach boolean networks bns are discrete time dynamical systems with boolean state variables and outputs bns are recently attracting considerable interest as computational models for genetic and cellular networks we consider the observability of bns that is the possibility of uniquely determining the initial state given a time sequence of outputs our main result is that determining whether a bn is observable is np hard this holds for both synchronous and asynchronous bns thus unless p np there does not exist an algorithm with polynomial time complexity that solves the observability problem we also give two simple algorithms with exponential complexity that solve this problem our results are based on combining the algebraic representation of bns derived by d cheng with a graph theoretic approach some of the theoretical results are applied to study the observability of a bn model of the mammalian cell cycle

finite element approximations in a non lipschitz domain part ii fil acosta gabriel universidad de buenos aires facultad de ciencias exactas y naturales departamento de matematica argentina

an hp finite element adaptive scheme to solve the laplace model for fluid solid vibrations abstract in this paper we introduce an hp finite element method to solve a two dimensional fluid structure spectral problem this problem arises from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid we prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues we define an a posteriori error estimator of the residual type which can be computed locally from the approximate eigenpair we show its reliability and efficiency by proving that the estimator is equivalent to the energy norm of the error up to higher order terms the equivalence constant of the efficiency estimate being suboptimal in that it depends on the polynomial degree we present an hp adaptive algorithm and several numerical tests which show the performance of the scheme including some numerical evidence of exponential convergence

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

connectivity properties of mandelbrot s percolation process in 1974 mandelbrot introduced a process in 0 1 2 which he called canonical curdling and later used in this book s on fractals to generate self similar random sets with hausdorff dimension d 0 2 in this paper we will study the connectivity or percolation properties of these sets proving all of the claims he made in sect 23 of the fractal geometry of nature and a new one that he did not anticipate there is a probability pc 0 1 so that if p pc then the set is duslike i e the largest connected component is a point whereas if p pc notice the opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles more succinctly put the system has a first order phase transition

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

on reducible state variables of logical control networks abstract reducible state variables are important for the study of reduced order observer and control design of logical control networks lcns this paper studies the number of reducible state variables of lcns based on the upper bound of reducible state variables several sufficient necessary conditions are presented for the number of reducible state variables in addition using the regular subspace theory the coordinate transformation as well as the swap matrix are constructed and some new criteria are proposed to calculate reducible state variables

random fractal strings their zeta functions complex dimensions and spectral asymptotics in this paper a string is a sequence of positive non increasing real numbers which sums to one for our purposes a fractal string is a string formed from the lengths of removed sub intervals created by a recursive decomposition of the unit interval by using the so called complex dimensions of the string the poles of an associated zeta function it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string we consider random versions of fractal strings we show that by using a random recursive self similar construction it is possible to obtain similar results to those for deterministic self similar strings in the case of strings generated by the excursions of stable subordinators we show that the complex dimensions can only lie on the real line the results allow us to discuss the geometric and spectral asymptotics of one dimensional domains with random fractal boundary

bi decomposition of multi valued logical functions and its applications abstract the bi decomposition of multi valued logical mvl functions including disjoint and non disjoint cases is considered using a semi tensor product an mvl function can be expressed in its algebraic form based on this form straightforward verifiable necessary and sufficient conditions are provided for each case respectively the constructive proofs also lead to constructing corresponding decompositions using these results the implicit function theorem ift of k valued functions as a special bi decomposition is obtained finally as an application the normalization of dynamic algebraic d a boolean networks is investigated using ift of k valued functions

a finite element solution of an added mass formulation for coupled fluid solid vibrations a finite element method to approximate the vibration modes of a structure in contact with an incompressible fluid is analyzed in this paper the effect of the fluid is taken into account by means of an added mass formulation which is one of the most usual procedures in engineering practice gravity waves on the free surface of the liquid are also considered in the model piecewise linear continuous elements are used to discretize the solid displacements the variables to compute the added mass terms and the vertical displacement of the free surface yielding a non conforming method for the spectral coupled problem error estimates are settled for approximate eigenfunctions and eigenfrequencies implementation issues are discussed and numerical experiments are reported in particular the method is compared with other numerical scheme based on a pure displacement formulation which has been recently analyzed

finite element approximations in a nonlipschitz domain in this paper we analyze the approximation by standard piecewise linear finite elements of a nonhomogeneous neumann problem in a cuspidal domain since the domain is not lipschitz many of the results on sobolev spaces which are fundamental in the usual error analysis do not apply therefore we need to work with weighted sobolev spaces and to develop some new theorems on traces and extensions we show that in the domain considered here suboptimal order can be obtained with quasi uniform meshes even when the exact solution is in h 2 and we prove that the optimal order with respect to the number of nodes can be recovered by using appropriate graded meshes

algebraic formulation and topological structure of boolean networks with state dependent delay abstract this paper investigates the algebraic formulation topological structure and set stability of boolean networks with state dependent delay sdd firstly using the algebraic state space representation assr method the dynamics of boolean networks with sdd is converted into an equivalent augmented system secondly based on the equivalent augmented system some necessary and sufficient conditions are presented to calculate fixed points and cycles of boolean networks with sdd thirdly it is proved that the set stability of boolean networks with sdd is equivalent to the set stability of augmented system and a necessary and sufficient condition is presented by constructing a kind of set stability matrix finally the obtained results are applied to the strategy consensus of networked evolutionary games with memories

the nature of the atomic surfaces of quasiperiodic self similar structures quasiperiodic self similar chains generated by substitutions i e deterministic concatenation rules and their diffraction spectra are analysed in a systematic fashion from the viewpoint of the superspace formalism a substitution acting on n objects generates quasiperiodic chains if and only if the associated substitution matrix fulfils two arithmetic conditions pisot property and unit determinant the structures thus obtained can be alternatively built as sections of periodic patterns in an n dimensional superspace which are regular repetitions of an atomic surface the authors derive a general algorithm to construct this atomic surface it is a compact set of the n 1 dimensional internal space which is a unit cell for a lattice of translations the atomic surface is nevertheless not necessarily connected and its boundary is generically an anisotropic self similar fractal the dimension db of this boundary is shown to govern the anomalously slow fall off of the intensities of bragg diffractions and therefore to influence physical properties

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

recursive self similarity for random trees random triangulations and brownian excursion recursive self similarity for a random object is the property of being decomposable into independent rescaled copies of the original object certain random combinatorial objects trees and triangulations possess approximate versions of recursive self similarity and then their continuous limits possess exact recursive self similarity in particular since the limit continuum random tree can be identified with brownian excursion we get a nonobvious recursive self similarity property for brownian excursion

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

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

controllability decomposition of dynamic algebraic boolean control networks two types of controllability of dynamic algebraic boolean control networks are first presented the necessary and sufficient conditions of such controllability are derived a similar concept of coo

branching random walk with exponentially decreasing steps and stochastically self similar measures we consider a branching random walk on r whose step size decreases by a fixed factor 0 1 2 the limit measure is almost surely a s absolutely continuous with respect to the lebesgue measure but for pisot 1 it is a s singular 2 for all a 5 1 2 the support of the measure is a s the closure of its interior 3 for pisot 1 the support of the measure is fractured it is a s disconnected and the components of the complement are not isolated on both sides

nonhomogeneous neumann problem for the poisson equation in domains with an external cusp abstract in this work we analyze the existence and regularity of the solution of a nonhomogeneous neumann problem for the poisson equation in a plane domain with an external cusp in order to prove that there exists a unique solution in h 1 using the lax milgram theorem we need to apply a trace theorem since is not a lipschitz domain the standard trace theorem for h 1 does not apply in fact the restriction of h 1 functions is not necessarily in l 2 so we introduce a trace theorem by using weighted sobolev norms in under appropriate assumptions we prove that the solution of our problem is in h 2 and we obtain an a priori estimate for the second derivatives of the solution

lacunarity of self similar and stochastically self similar sets let k be a self similar set in rdd of hausdorff dimension d and denote by ik c l the d dimensional lebesgue measure of its 6 neighborhood we study the limiting behavior of the quantity e d d ik e i as 6 e 0 it turns out that this quantity does not have a limit in many interesting cases including the usual ternary cantor set and the sierpinski carpet we also study the above asymptotics for stochastically self similar sets the latter results then apply to zero sets of stable bridges which are stochastically self similar in the sense of the present paper and then more generally to level sets of stable processes specifically it follows that if kt is the zero set of a realvalued stable process of index a c 1 2 run up to time t then e siit e i converges to a constant multiple of the local time at 0 simultaneously for all t 0 on a set of probability one the asymptotics for deterministic sets are obtained via the renewal theorem the renewal theorem also yields asymptotics for the mean 1e ik c in the random case while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks

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

self similar random measures are locally scale invariant in an earlier paper patzschke and u zahle 11 have proved the existence of a fractional tangent measure at the typical point of a self similar random measure under rather special technical assumptions in the present paper we remove the most restrictive one here we suppose the open set condition for the similarities a constant positive lower bound for the random contraction ratios and vanishing on the boundary of the open set with probability 1 the tangent measure isd scale invariant whered is the similarity dimension of moreover we approximate the tangential distribution by means of and use this in order to prove that the hausdorff dimension of the tangent measure equalsd since the former coincides with the hausdorff dimension of we obtain an earlier result of mauldin and williams 9 as a corollary

elliptic equations with nonzero boundary conditions in weighted sobolev spaces abstract we present weighted sobolev spaces h p along with a trace theorem and an interpolation theorem for the spaces then we solve nonzero boundary value problems for elliptic equations in h p

strategy optimization for static games based on stp method abstract in this paper we consider the choice behavior of players in a static game and discuss solutions to static games of complete information problem via semi tensor product stp method some properties of the dominant behaviors are obtained based on which choosing the dominant strategies deleting the dominated strategies are used to optimize the strategy sets so as to obtain solutions to the considered game examples are given to show the effectiveness of the results

self avoiding process on the sierpinski gasket we construct a self avoiding process taking values in the finite sierpinski gasket and study its properties we then study continuum limit processes that are suggested by the statistical mechanics of self avoiding paths on the pre sierpinski gasket we prove that there are three types of continuum limit processes according to the parameters defining the statistical mechanics of self avoiding paths i the self avoiding process we construct in this paper ii a deterministic motion along a peano curve on the finite sierpinski gasket iii a deterministic motion along a line segment

on the asymptotics of the eigenvalue counting function for random recursive sierpinski gaskets we consider natural laplace operators on random recursive affine nested fractals based on the sierpinski gasket and prove an analogue of weyl s classical result on their eigenvalue asymptotics the eigenvalue counting function n is shown to be of order ds 2 as where we can explicitly compute the spectral dimension d s moreover the limit n ds 2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable this random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal