upper critical dimension of the kardar parisi zhang equation numerical results for the directed polymer model in 1 4 dimensions in various types of disorder are presented the results are obtained for a system size that is considerably larger than considered previously for the extreme strong disorder case min max system associated with the directed percolation model the expected value of the meandering exponent 0 5 is clearly revealed with very weak finite size effects for the weak disorder case associated with the kardar parisi zhang equation finite size effects are stronger but the value of is clearly seen in the vicinity of 0 57 in systems with strong disorder it is expected that the system will cross over sharply from min max behavior at short chains to weak disorder behavior at long chains our numerical results agree with that expectation to complete the picture we obtain the energy fluctuation exponent for weak disorder and we find that the value of is in the vicinity of 0 14 thus the meandering exponent and the energy fluctuation exponent obey the strong coupling scaling relation 2 1 our results indicate that 1 4 is not the upper critical dimension in the weak disorder case and thus 4 1 does not seem to be the upper critical dimension for the kardar parisi zhang equation

on saint venant s principle in finite anti plane shear an energy approach some time ago a version of saint venant s principle was formulated and established for finite elastostatics 1 as was discussed in 1 the issues of concern in connection with saint venant s principle in the nonlinear theory of elasticity are considerably more involved then those arising in the linear theory for a survey of results on saint venant s principle primarily for linear theories see e g 2 5 one difficulty is that the appropriate saint venant solutions need to be carefully characterized see e g 6 13 and the references cited therein secondly in the absence of superposition consideration of self equilibrated end loads is no longer sufficient furthermore instabilities may have to be taken into account also the decay rate for end effects even if exponential might depend on the overall loading as well as on geometry and material characteristics several of these issues have been considered in recent studies in the nonlinear elasticity context 1 14 23 as well as in investigations of spatial decay of solutions of nonlinear elliptic partial differential equations 24 32

numerical and theoretical studies of noise effects in the kauffman model in this work we analyze the stochastic dynamics of the kauffman model evolving under the influence of noise by considering the average crossing time between two distinct trajectories we show that different kauffman models exhibit a similar kind of behavior even when the structure of their basins of attraction is quite different this can be considered as a robust property of these models we present numerical results for the full range of noise level and obtain approximate analytic expressions for the above crossing time as a function of the noise in the limit cases of small and large noise levels

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

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

decay estimates for a class of second order quasilinear equations in three dimensions in a recent paper 1 the authors made use of differential inequality techniques to derive exponential decay estimates for a class of second order quasilinear equations not necessarily elliptic in the plane specifically they were concerned with solutions of equations in divergence form defined in a semi infinite strip with non zero boundary data on one end only decay estimates in various norms were derived

complexity of the predecessor problem in kauffman networks kauffman nets also known as n k models have been studied extensively because their dynamics can be used to model a variety of interesting dynamical processes this paper investigates the properties of the problem of determining whether or not a given configuration of a kauffman net has a predecessor here it is shown that when the parameter k that governs the number of connections grows as ln n where n is the number of elements the problem of finding a solution is extremely sensitive to small changes in the problem statement this result has implications for studies of the physics of random systems and also may have applications for questions in computational complexity theory

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

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

on the spatial decay of the energy for some quasilinear boundary value problems in semi infinite cylinders dans un cylindre semi infini a 3 dimensions r on considere le probleme p x u u u j j 0 avec des donnees aux limites de type dirichlet ou neumann on deduit une borne d energie

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

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

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

nonperturbative renormalization group for the stationary kardar parisi zhang equation scaling functions and amplitude ratios in 1 1 2 1 and 3 1 dimensions we investigate the strong coupling regime of the stationary kardar parisi zhang equation for interfaces growing on a substrate of dimension d 1 2 and 3 using a nonperturbative renormalization group nprg approach we compute critical exponents correlation and response functions extract the related scaling functions and calculate universal amplitude ratios we work with a simplified implementation of the second order in the response field approximation proposed in a previous work phys rev e 84 061150 2011 and phys rev e 86 019904 e 2012 which greatly simplifies the frequency sector of the nprg flow equations while keeping a nontrivial frequency dependence for the two point functions the one dimensional scaling function obtained within this approach compares very accurately with the scaling function obtained from the full second order nprg equations and with the exact scaling function furthermore the approach is easily applicable to higher dimensions and we provide scaling functions and amplitude ratios in d 2 and d 3 we argue that our ansatz is reliable up to d symbol see text 3 5

density matrix formulation for quantum renormalization groups a generalization of the numerical renormalization group procedure used first by wilson for the kondo problem is presented it is shown that this formulation is optimal in a certain sense as a demonstration of the effectiveness of this approach results from numerical real space renormalization group calculations for heisenberg chains are presented

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

exponential decay estimates for solutions of the von karman equations on a semi infinite strip this paper is concerned with the analysis of saint venant edge effects for nonlinear elastic plates the model used is based on the von karman plate equations a coupled system of two nonlinear elliptic partial differential equations with the biharmonic operator as the principal part energy methods are used to establish a nonlinear integro differential inequality for a quadratic functional arguments based on comparison theorems are then used to establish exponential decay of end effects