stability of boolean networks the joint effects of topology and update rules we study the stability of orbits in large boolean networks we treat the case in which the network has a given complex topology and we do not assume a specific form for the update rules which may be correlated with local topological properties of the network while recent past work has addressed the separate effects of complex network topology and certain classes of update rules on stability only crude results exist about how these effects interact we present a widely applicable solution to this problem numerical simulations confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

power law distributions in some random boolean networks the kauffman net is a dynamical system of logical variables receiving two random inputs and each randomly assigned a boolean function we show that the attractor and transient lengths exhibit scaleless behavior with power law distributions over up to 10 orders of magnitude in probability our results provide evidence for the existence of the edge of chaos as a distinct regime between the ordered and chaotic phases analogous to a critical point in statistical mechanics the power law distributions are robust to the changes in the composition of the transition rules and network dynamics

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

the effect of network topology on the stability of discrete state models of genetic control boolean networks have been proposed as potentially useful models for genetic control an important aspect of these networks is the stability of their dynamics in response to small perturbations previous approaches to stability have assumed uncorrelated random network structure real gene networks typically have nontrivial topology significantly different from the random network paradigm to address such situations we present a general method for determining the stability of large boolean networks of any specified network topology and predicting their steady state behavior in response to small perturbations additionally we generalize to the case where individual genes have a distribution of expression biases and we consider a nonsynchronous update as well as extension of our method to non boolean models in which there are more than two possible gene states we find that stability is governed by the maximum eigenvalue of a modified adjacency matrix and we test this result by comparison with numerical simulations we also discuss the possible application of our work to experimentally inferred gene networks

invertible cellular automata a review abstract in the light of recent developments in the theory of invertible cellular automata we attempt to give a unified presentation of the subject and discuss its relevance to computer science and mathematical physics

uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods assumed stress hybrid methods are known to improve the performance of standard displacement based finite elements and are widely used in computational mechanics the methods are based on the hellinger reissner variational principle for the displacement and stress variables this work analyzes two existing 4 node hybrid stress quadrilateral elements due to pian and sumihara t h h pian k sumihara rational approach for assumed stress finite elements int j numer methods engrg 20 9 1984 1685 1695 and due to xie and zhou x p xie t x zhou optimization of stress modes by energy compatibility for 4 node hybrid quadrilaterals int j numer methods engrg 59 2004 293 313 which behave robustly in numerical benchmark tests for the finite elements the isoparametric bilinear interpolation is used for the displacement approximation while different piecewise independent 5 parameter modes are employed for the stress approximation we show that the two schemes are free from poisson locking in the sense that the error bound in the a priori estimate is independent of the relevant lame constant we also establish the equivalence of the methods to two assumed enhanced strain schemes finally we derive reliable and efficient residual based a posteriori error estimators for the stress in l2 norm and the displacement in h1 norm and verify the theoretical results by some numerical experiments

median attractor and transients in random boolean nets abstract variants of a random boolean network of binary gates called the kauffman net are studied each gate receives two random inputs from the other gates in the network and is assigned a transition function from a non uniform distribution fine upper bounds on the attractor and transient lengths as the behavioral characteristics of the network are determined for large net size n and different transition function distributions the effects of these distributions on the localization property of the net is shown analysis of nets with n 1000 was shown to be very difficult we also determine the scaling properties of attractors and transients with the net size our results exhibit strong antichaos behavior even for very large n for some of the networks

using the renormalization group to classify boolean functions this paper argues that the renormalization group technique used to characterize phase transitions in condensed matter systems can be used to classify boolean functions a renormalization group transformation is presented that maps an arbitrary boolean function of n boolean variables to one of n 1 variables applying this transformation to a generic boolean function one whose output for each input is chosen randomly and independently to be one or zero with equal probability yields another generic boolean function moreover applying the transformation to some other functions known to be non generic such as boolean functions that can be written as polynomials of degree with n and functions that depend on composite variables such as the arithmetic sum of the inputs yields non generic results one can thus define different phases of boolean functions as classes of functions with different types of behavior upon repeated application of the renormalization transformation possible relationships between different phases of boolean functions and computational complexity classes studied in computer science are discussed

a two dimensional modified levy walk model for the dna sequences abstract a two dimensional modified levy walk model for the dna sequences with only three parameters is proposed this model is applied to simulate real dna sequence data of the saccharomyces cerevisiae genome dna sequences are converted into one and two dimensional random walk mappings for statistical study satisfactory statistical results are obtained when 16 chromosomes in the genome are all analyzed and compared to the model sequence data measurements on the root mean square deviations are in good agreement when real sequences and model sequences are compared levy type statistics plays a major role in generating long range correlation phenomena power values 0 64 0 68 are obtained for the power law f l l where f l is the root mean square deviation at step size l the same model is applied to sequences of some other species fair agreement is also obtained

reversible boolean networks i distribution of cycle lengths abstract we consider a class of models describing the dynamics of n boolean variables where the time evolution of each depends on the values of k of the other variables previous work has considered models with dissipative dynamics here we consider time reversible models which necessarily have the property that every possible point in the state space is an element of one and only one cycle the orbits can be classified by their behavior under time reversal the orbits that transform into themselves under time reversal have properties quite different from those that do not in particular a significant fraction of latter type orbits have lengths enormously longer than orbits that are time reversal symmetric for large k and moderate n the vast majority of points in the state space are on one of the time reversal singlet orbits and a random hopping model gives an accurate description of orbit lengths however for any finite k the random hopping approximation fails qualitatively when n is large enough n 22k as in the dissipative case when k is large typical orbit lengths grow exponentially with n whereas for small enough k typical orbit lengths grow much more slowly with n the numerical data are consistent with the existence of a phase transition at which the average orbit length grows as a power of n at a value of k between 1 4 and 1 7 however in the reversible models the interplay between the discrete symmetry and quenched randomness can lead to enormous fluctuations of orbit lengths and other interesting features that are unique to the reversible case

exhaustive percolation on random networks we consider propagation models that describe the spreading of an attribute called damage through the nodes of a random network in some systems the average fraction of nodes that remain undamaged vanishes in the large system limit a phenomenon we refer to as exhaustive percolation we derive scaling law exponents and exact results for the distribution of the number of undamaged nodes valid for a broad class of random networks at the exhaustive percolation transition and in the exhaustive percolation regime this class includes processes that determine the set of frozen nodes in random boolean networks with indegree distributions that decay sufficiently rapidly with the number of inputs connections between our calculational methods and previous studies of percolation beginning from a single initial node are also pointed out central to our approach is the observation that key aspects of damage spreading on a random network are fully characterized by a single function specifying the probability that a given node will be damaged as a function of the fraction of damaged nodes in addition to our analytical investigations of random networks we present a numerical example of exhaustive percolation on a directed lattice less

antichaos in a class of random boolean cellular automata abstract a variant of kauffman s model of cellular metabolism is presented it is a randomly generated network of boolean gates identical to kauffman s except for a small bias in favor of boolean gates that depend on at most one input the bias is asymptotic to 0 as the number of gates increases upper bounds on the time until the network reaches a state cycle and the size of the state cycle as functions of the number of gates n are derived for any c 0 if the bias approaches 0 slowly enough the state cycles will be smaller than nc

generalized variability response functions for two dimensional elasticity problems the ability to determine probabilistic characteristics of response quantities in structural mechanics e g displacements stresses as well as effective material properties is restricted due to lack of information on the probabilistic characteristics of the uncertain system parameters the concept of the variability response function vrf has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous random fields on the uncertain structural response the key property of the vrf in its classical sense is its independence from the marginal probability distribution function pdf and the spectral density function sdf of the uncertain system parameters it depends only on the deterministic structural configuration and boundary conditions proofs have been provided for the existence of vrfs for linear and some nonlinear statically determinate beams for statically indeterminate structures the monte carlo based generalized variability response function gvrf methodology has been proposed recently as a generalization of the vrf concept to indeterminate linear and some nonlinear beams the methodology computes gvrfs which are analogous to vrfs for statically determinate structures and evaluates their dependence or lack thereof on the pdf and sdf of the random field thereby providing an estimate of the accuracy of the gvrf in this paper the gvrf methodology is extended to problems involving two dimensional linear continua whose stochasticity is characterized by statistically homogeneous random fields after detailing the gvrf methodology for two dimensional random fields two numerical examples are provided gvrfs are computed for the displacement response and for the effective compliance of linear plane stress systems

scale free network with boolean dynamics as a function of connectivity in this work we analyze scale free networks with different power law spectra n k ensuremath sim k ensuremath ensuremath gamma under a boolean dynamic where the boolean rule that each node obeys is a function of its connectivity k this is done by using only two logical functions and and xor which are controlled by a parameter q using a damage spreading technique we show that the hamming distance and the number of 1 s exhibit power law behavior as a function of q the exponents appearing in the power laws depend on the value of ensuremath gamma

relevant elements magnetization and dynamical properties in kauffman networks a numerical study abstract this is the first of two papers about the structure of kauffman networks in this paper we define the relevant elements of random networks of automata following previous work by flyvbjerg j phys a 21 1988 l955 l960 and flyvbjerg and kjaer j phys a 21 1988 1695 1718 and we study numerically their probability distributions in the chaotic phase and on the critical line of the model a simple approximate argument predicts that their number scales as n on the critical line while it is linear with n in the chaotic phase and independent on system size in the frozen phase this argument is confirmed by numerical results the study of the relevant elements gives useful information about the properties of the attractors in critical networks where the pictures coming from either approximate computation methods or from simulations are not very clear

emergent properties in random complex automata abstract studies of large randomly assembled binary boolean node automata have demonstrated that such systems can spontaneously exhibit enormously ordered dynamical behavior an important approach to characterizing these behaviors has consisted of studying ensembles of automata ensemble specifications have been based on 1 choice of boolean function regulating each node in the automaton 2 random or biased mappings of the 2 n automata states into themselves 3 the numbers of inputs k per node this article briefly reviews these approaches reports new results on the expected behavior of threshold automata and automata with a particular class of biased mappings in addition it discusses emergent highly ordered dynamical behavior in random automata rich in a specific class of canalizing boolean functions due to the crystallization of powerful subautomata called forcing structures the ordered behaviors may be of importance in biological evolution in physics and in the design of adaptive automata

lyapunov exponents in random boolean networks a new order parameter approximation to random boolean networks rbn is introduced based on the concept of boolean derivative a statistical argument involving an annealed approximation is used allowing to measure the order parameter in terms of the statistical properties of a random matrix using the same formalism a lyapunov exponent is calculated allowing to provide the onset of damage spreading through the network and how sensitive it is to minimal perturbations finally the lyapunov exponents are obtained by means of dierent approximations through distance method and a discrete variant of the wolf s method for continuous systems c 2000 elsevier science b v all rights reserved

multifractality and the kauffman model the author simulates the spread of an initial damage in the kauffman cellular automata at its critical point on the square lattice he calculates various moments of the probability that a site has been damaged n times and checks for multifractality in the critical exponents for different ensembles specifically he finds no evidence for multifractality when the moments of the probabilities are evaluated with lattice size l but multifractal behaviour occurs when the moments are monitored as a function of time t

strong convergence of mann s type iterations for nonexpansive semigroups in general banach spaces abstract in this paper we prove a strong convergence theorem of mann s type for commutative nonexpansive semigroups in general banach spaces using this theorem we obtain some strong convergence theorems in general banach spaces

an ergodic theorem for semigroups of nonexpansive mappings in a hilbert space abstract j b baillon c r acad sci paris ser a 280 1975 1511 1514 proved an ergodic theorem for a single nonexpansive mapping in a hilbert space which is a nonlinear version of von neumann s mean ergodic theorem in this paper we study the ergodic behavior of a semigroup of nonexpansive mappings we try to find a sequence of means on the semigroup generalizing the cesaro means on n such that the corresponding sequence of nonexpansive mappings converges to a projection onto the set of common fixed points our method of proof is an appropriate modification of a pazy s proof israel j math 26 1977 197 204 of baillon s theorem

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

stochastic collocation methods on unstructured grids in high dimensions via interpolation in this paper we propose a method for conducting stochastic collocation on arbitrary sets of nodes to accomplish this we present the framework of least orthogonal interpolation which allows one to construct interpolation polynomials based on arbitrarily located grids in arbitrary dimensions these interpolation polynomials are constructed as a subspace of the family of orthogonal polynomials corresponding to the probability distribution function on stochastic space this feature enables one to conduct stochastic collocation simulations in practical problems where one cannot adopt some popular node selections such as sparse grids or cubature nodes we present in detail both the mathematical properties of the least orthogonal interpolation and its practical implementation algorithm numerical benchmark problems are also presented to demonstrate the efficacy of the method

phase transition for the threshold contact process an approximation of heterogeneous random boolean networks we consider a model for heterogeneous gene regulatory networks that is a generalization of the model proposed by chatterjee and durrett random struct algorithms 39 2 228 246 2011 as an annealed approximation of kauffmann s j theor biol 22 3 437 467 1969 random boolean networks in this model genes are represented by the vertices of a random directed graph mathscr g n on n vertices having specified in degree distribution mathbf p text in resp joint distribution mathbf p text in out of in degree and out degree and the expression bias the expected fraction of 1 s in the boolean functions p is same for all vertices following chatterjee and durrett random struct algorithms 39 2 228 246 2011 and a standard practice in the physics literature we use a discrete time threshold contact process with parameter q 2p 1 p in which vertices are either occupied or vacant a vertex with at least one resp no occupied input at time t will be occupied at time t 1 with probability q resp 0 on mathscr g n to approximate the dynamics of the boolean network we find a parameter r which has an explicit expression in terms of certain moments of mathbf p text in resp mathbf p text in out such that with probability tending to 1 as n goes to infinity if r cdot 2p 1 p 1 then starting from all occupied sites the threshold contact process on mathscr g n maintains a positive quasi stationary density pi mathbf p text in resp pi mathbf p text in out of occupied sites till time which is exponential in n whereas if r cdot 2p 1 p 1 then the persistence time of the threshold contact process on mathscr g n is at most logarithmic in n these two phases correspond to the chaotic and ordered behavior of the gene networks respectively

distribution of local magnetisations in random networks of automata the authors derive an integral equation which gives the distribution p m of local magnetisations for a random network of automata kauffman model they show how from this distribution one can recover the distances between configurations and the fraction of spins in the stable core i e the spins which do not depend on the initial conditions a numerical solution of the integral equation shows that p m has singularities at m 0 or 1 2 and or 1

broad edge of chaos in strongly heterogeneous boolean networks the dynamic stability of the boolean networks representing a model for the gene transcriptional regulation kauffman model is studied by calculating analytically and numerically the hamming distance between two evolving configurations this turns out to behave in a universal way close to the phase boundary only for in degree distributions with a finite second moment in degree distributions of the form pd k k with 2 3 thus having a diverging second moment lead to a slower increase of the hamming distance when moving towards the unstable phase and to a broadening of the phase boundary for finite n with decreasing we conclude that the heterogeneous regulatory network connectivity facilitates the balancing between robustness and evolvability in living organisms

binary threshold networks as a natural null model for biological networks spin models of neural networks and genetic networks are considered elegant as they are accessible to statistical mechanics tools for spin glasses and magnetic systems however the conventional choice of variables in spin systems may cause problems in some models when parameter choices are unrealistic from a biological perspective obviously this may limit the role of a model as a template model for biological systems perhaps less obviously also ensembles of random networks are affected and may exhibit different critical properties we consider here a prototypical network model that is biologically plausible in its local mechanisms we study a discrete dynamical network with two characteristic properties nodes with binary states 0 and 1 and a modified threshold function with ensuremath theta 0 0 0 we explore the critical properties of random networks of such nodes and find a critical connectivity k c 2 0 with activity vanishing at the critical point finally we observe that the present model allows a more natural implementation of recent models of budding yeast and fission yeast cell cycle control networks

closing probabilities in the kauffman model an annealed computation abstract we use a probabilistic approach to compute the distribution of periods transients and weights of attraction basins in kauffman networks the results for these quantities are obtained in the framework of the annealed approximation first introduced by derrida and pomeau numerical results for the average periods are in good agreement with the computed values of the exponents we report also on some interesting features which cannot be explained within the annealed approximation

optimization of stress modes by energy compatibility for 4 node hybrid quadrilaterals optimal hybrid stress quadrilaterals can be obtained by adopting appropriate stresses and displacements and satisfying the energy compatibility condition is shown to be an ultimate key to obtaining optimal stress modes by using compatible isoparametric bilinear q4 displacements and 5 parameter energy compatible stresses of the combined hybrid finite element ch 0 1 a robust 4 node plane stress element ecq4 is derived equivalence to another hybrid stress element lq6 with 9 parameter complete linear stresses based on a modified hellinger reissner principle is established a convergence analysis is given and numerical experiments show that elements ecq4 lq6 have high performance i e are accurate at coarse meshes insensitive to mesh distortions and free from locking copyright 2003 john wiley sons ltd

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

generalized polynomial chaos and random oscillators we present a new approach to obtain solutions for general random oscillators using a broad class of polynomial chaos expansions which are more efficient than the classical wiener hermite expansions the approach is general but here we present results for linear oscillators only with random forcing or random coefficients in this context we are able to obtain relatively sharp error estimates in the representation of the stochastic input as well as the solution we have also performed computational comparisons with monte carlo simulations which show that the new approach can be orders of magnitude faster especially for compact distributions

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

relaxation closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks attractors in asymmetric neural networks with deterministic parallel dynamics present a chaotic regime at symmetry where the average length of the cycles increases exponentially with system size and an oscillatory regime at high symmetry where the average length of the cycles is 2 we show both with analytic arguments and numerically that there is a sharp transition at a critical symmetry between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size and the weights of their attraction basins are distributed as in a random map with reversal symmetry the timescale after which cycles are reached grows exponentially with system size n and the exponent vanishes in the symmetric limit where the transition can be related to the dynamics of the infinite system where cycles are never reached using the closing probabilities as a tool we also study the relaxation of the function where is the local field experienced by the neuron i in the symmetric system it plays the role of a ljapunov function which drives the system towards its minima through steepest descent this interpretation survives even if only on the average also for small asymmetry this acts like an effective temperature the larger the asymmetry the faster the relaxation and the higher the asymptotic value reached e reaches very deep minima in the fixed points of the dynamics which are reached with vanishing probability and attains a larger value on the typical attractors which are cycles of length 2

criticality in random threshold networks annealed approximation and beyond random threshold networks with sparse asymmetric connections show complex dynamical behavior similar to random boolean networks with a transition from ordered to chaotic dynamics at a critical average connectivity kc in this type of model contrary to boolean networks propagation of local perturbations damage depends on the in degree of the sites kc is d eterminedanalytically using an annealedapproximation andthe results are con2rmedby numerical simulations it is shown that the statistical distributions of damage spreading near the percolation transition obey power laws andd ynamical correlations between active network clusters become maximal we investigate the e4ect of local damage suppression at highly connected nodes for networks with scale free in degree distributions possible relations of our 2ndings to properties of real world networks like robustness and non trivial degree distributions are discussed c

kauffman boolean model in undirected scale free networks we investigate analytically and numerically the critical line in undirected random boolean networks with arbitrary degree distributions including the scale free topology of connections p k ensuremath sim k ensuremath ensuremath gamma we explain that the unattainability of the critical line in numerical simulations of classical random graphs is due to percolation phenomena we suggest that recent findings of discrepancy between simulations and theory in directed random boolean networks might have the same reason we also show that in infinite scale free networks the transition between frozen and chaotic phases occurs for 3l ensuremath gamma l3 5 since most critical phenomena in scale free networks reveal their nontrivial character for ensuremath gamma l3 the position of the critical line in the kauffman model seems to be an important exception to the rule

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

analysis of some quadrilateral nonconforming elements for incompressible elasticity in this work some nonconforming elements on arbitrary quadrilateral meshes in solving incompressible elastic equations are analyzed a uniform optimal convergence rate is established at the incompressible limit nu 0 5 for both displacement and stresses or pressure in the case of incompressible flow

length of state cycles of random boolean networks an analytic study in this paper we consider the mean length of transients and the length of state cycles in random boolean networks we present an approximate calculation of these quantities as a function of the size and connectivity of the network where using an annealed approximation we derive a recursive formula for the length of steps of the system using the mean step length and an effective momentary state space we calculate an approximate formula for the probability distribution function pdf of the length of state cycles and transients we compare this pdf with analytical results in special cases and with simulations by the monte carlo procedure

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

deducing local rules for solving global tasks with random boolean networks abstract it has been shown that uniform as well as non uniform cellular automata ca can be evolved to perform certain computational tasks random boolean networks are a generalization of two state cellular automata where the interconnection topology and the cell s rules are specified at random here we present a novel analytical approach to find the local rules of random boolean networks rbns to solve the global density classification and the synchronization task from any initial configuration we quantitatively and qualitatively compare our results with previously published work on cellular automata and show that randomly interconnected automata are computationally more efficient in solving these two global tasks our approach also provides convergence and quality estimates and allows the networks to be randomly rewired during operation without affecting the global performance finally we show that rbns outperform small world topologies on the density classification task and that they perform equally well on the synchronization task our novel approach and the results may have applications in designing robust complex networks and locally interacting distributed computing systems for solving global tasks

a percolation approach to the kauffman model a study is made of the spreading of damage in the random but deterministic kauffman model on the square lattice with the spreading from one edge of the lattice the critical value of the parameterp c above which the system becomes chaotic is found to bep c 0 298 the possibility of suppression of the chaotic phase by noise is also studied it is found that forp p c an extremely large noise levelg 0 99 is required if possible at all

the modular structure of kauffman networks this is the second paper of a series of two about the structural properties that influence the asymptotic dynamics of random boolean networks here we study the functionally independent clusters in which the relevant elements introduced and studied in our first paper 3 are subdivided we show that the phase transition in random boolean networks can also be described as a percolation transition the statistical properties of the clusters of relevant elements that we call modules give an insight on the scaling behavior of the attractors of the critical networks that according to kauffman have a biological analogy as a model of genetic regulatory systems

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

convergence theorems of fixed points for lipschitz pseudo contractions in hilbert spaces abstract let c be a closed convex subset of a real hilbert space h and assume that t is a strict pseudo contraction on c consider mann s iteration algorithm given by x 0 c x n 1 n x n 1 n t x n n 0 it is proved that if the control sequence n is chosen so that n 1 and n 0 n 1 n then lim n x n t x n d 0 r a where a i t and d 0 d denotes the distance between the origin and the subset set d of h as a consequence of this result we prove that if t has a fixed point in c then x n converges weakly to a fixed point of t also we extend a result due to reich to strict pseudo contractions in the hilbert space setting further by virtue of hybridization projections we establish a strong convergence theorem for lipschitz pseudo contractions the results presented in this paper improve or extend the corresponding results of browder and petryshyn f e browder w v petryshyn construction of fixed points of nonlinear mappings in hilbert spaces j math anal appl 20 1967 197 228 rhoades b e rhoades fixed point iterations using infinite matrices trans amer math soc 196 1974 162 176 and of marino and xu g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 1 2007 336 346

critical dynamics of randomly assembled and diluted threshold networks the dynamical behavior of a class of randomly assembled networks of binary threshold units subject to random deletion of connections is studied based on the annealed approximation suitable in the thermodynamic limit the dynamical phase diagram is constructed for several forms of the probability density distribution of nonvanishing connection strengths the family of power law distribution functions rho0 x 1 alpha 2 x alpha is found to play a special role in expanding the domain of stable ordered dynamics at the expense of the disordered chaotic phase relationships with other recent studies of the dynamics of complex networks allowing for variable in degree of the units are explored the relevance of the pruning of network connections to neural modeling and developmental neurobiology is discussed

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it treats easily a wide range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

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

order disorder phase transition in random walk networks in this paper we study in detail the behavior of random walk networks rwn s these networks are a generalization of the well known random boolean networks rbn s a classical approach to the study of the genome rwn s are also discrete networks but their response is defined by small variations in the state of each gene thus being a more realistic representation of the genome and a natural bridge between discrete and continuous models rwn s show a clear transition between order and disorder here we explicitly deduce the formula of the critical line for the annealed model and compute numerically the transition points for quenched and annealed models we show that rbn s and the annealed model of rwn s act as an upper and a lower limit for the quenched model of rwn s finally we calculate the limit of the annealed model for the continuous case

random boolean networks response to external periodic signals random boolean networks have been proposed as discrete models of genetic networks depending on the values of their control parameters these networks fall by themselves in order or disorder phases these networks are autonomous systems no external inputs are considered nevertheless in the real world the genetic networks are in5uenced by external signals many biological rhythms have 24 h periods related to sunlight coupled with molecular clocks in this work we study the response of random boolean networks to analytical and non analytical external periodic signals the relationship between external and internal parameters for the determination of the dynamical behaviour of this networks is theoretically determined and simulations in agreement with these theoretical results are reported c

multivalley structure in kauffman s model analogy with spin glasses kauffman s model describes a random network of automata the authors calculations indicate that the multivalley structure of the basins of attraction in kauffman s model is very similar to that of infinite range spin glasses the similarity with spin glasses is tested quantitatively by computing the probability that two initial configurations fall into the same valley

the number and probability of canalizing functions canalizing functions have important applications in physics and biology for example they represent a mechanism capable of stabilizing chaotic behavior in boolean network models of discrete dynamical systems when comparing the class of canalizing functions to other classes of functions with respect to their evolutionary plausibility as emergent control rules in genetic regulatory systems it is informative to know the number of canalizing functions with a given number of input variables this is also important in the context of using the class of canalizing functions as a constraint during the inference of genetic networks from gene expression data to this end we derive an exact formula for the number of canalizing boolean functions of n variables we also derive a formula for the probability that a random boolean function is canalizing for any given bias p of taking the value 1 in addition we consider the number and probability of boolean functions that are canalizing for exactly k variables finally we provide an algorithm for randomly generating canalizing functions with a given bias p and any number of variables which is needed for monte carlo simulations of boolean networks

reversible boolean networks ii phase transitions oscillations and local structures we continue our consideration of a class of models describing the reversible dynamics of n boolean variables each with k inputs we investigate in detail the behavior of the hamming distance as well as of the distribution of orbit lengths as n and k are varied we present numerical evidence for a phase transition in the behavior of the hamming distance at a critical value kc 1 62 and also an analytic theory that yields the exact bounds 1 5 k c 2 we also discuss the large oscillations that we observe in the hamming distance for k k c as a function of time as well as in the distribution of cycle lengths as a function of cycle length for moderate k both greater than and less than kc we propose that local structures or subsets of spins whose dynamics are not fully coupled to the other spins in the system play a crucial role in generating these oscillations the simplest of these structures are linear chains called linkages and rings called circuits we discuss the properties of the linkages in some detail and sketch the properties of circuits we argue that the observed oscillation phenomena can be largely understood in terms of these local structures 2001 published by elsevier science b v

a no go theorem for the existence of a discrete action principle abstract in this paper we study the problem of the existence of a least action principle for invertible second order dynamical systems discrete in time and space we show that when the configuration space is finite and arbitrary state transitions are allowed a least action principle does not exist for such systems we dichotomize discrete dynamical systems with infinite configuration spaces into those of finite type for which this theorem continues to hold and those not of finite type for which it is possible to construct a least action principle we also show how to recover an action by restriction of the phase space of certain second order discrete dynamical systems we provide numerous examples to illustrate each of these results

nonexpansive retractions and nonlinear ergodic theorems in banach spaces on donne des theoremes ergodiques non lineaires pour des semigroupes commutatifs d applications non dilatation sans utiliser le concept de moyenne compacte

random walk networks abstract random boolean networks are among the best known systems used to model genetic networks they show an on off dynamics and it is easy to obtain analytical results with them unfortunately very few genes are strictly on off switched on the other hand continuous methods are in principle more suitable to capture the real behavior of the genome but have difficulties when trying to obtain analytical results in this work we introduce a new model of random discrete network random walk networks where the state of each gene is changed by small discrete variations being thus a natural bridge between discrete and continuous models

an order parameter for networks of automata an exact polynomial equation is given for the size of the stable core of networks of automata with random connections when the connectivity k of a network equals 1 2 3 4 or 5 this equation is exactly solvable it is found that the size of the stable core is an order parameter for a phase transition well known from kauffman s model a new derivation of critical parameter values follows the phase structure is found to be independent of the updating scheme used in the dynamical law for the network

a unified analysis for stress strain hybrid methods of high performance abstract several methods have been developed in the literatures of computational mechanics to improve the performance of the conventional lower order displacement finite elements which yield poor results for problems with bending and for nearly incompressible medium this paper is devoted to a unified analysis of convergence for pian sumihara s chen cheung s and piltner taylor s enhanced stress strain schemes by virtue of the energy compatibility and the rank condition error estimates for these typical finite elements of high performance are obtained in a unified framework and especially weakly locking free error estimates with respect to the poisson s ratio in energy norms are obtained uniformly for 1 ch 2 as h 0 where c is a constant independent of and the mesh size h very much the same about the three methods is pointed out

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

some remarks on the set of common fixed points of one parameter semigroups of nonexpansive mappings in banach spaces with the opial property abstract in this paper we discuss the necessary and sufficient condition for common fixed points of one parameter nonexpansive semigroups in banach spaces with the opial property we next prove the convergence theorem to a common fixed point finally we discuss the existence of nonexpansive retraction onto the set of common fixed points in these theorems we may not assume that the strict uniform convexity of a banach space

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

transients in a one dimensional probabilistic cellular automaton abstract the characteristics of the distribution of transient times of a one dimensional probabilistic cellular automaton are studied by computer simulation the mean and width of the distribution are found to diverge by the same power law at each of the two critical points p c 1 0 and p c 2 0 75 of the model critical exponents obtained from finite size scaling indicate that p c 1 belongs to the universality class of directed random walks whereas p c 2 belongs to the universality class of directed percolation between the two critical points there exists a point of minimum transient length at p m 0 23 where the mean transient time scales logarithmically with the system size

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