slow motion manifolds far from the attractor in multistable reaction diffusion equations abstract we consider a scalar reaction diffusion equation with multistable nonlinearity with a particular symmetry by reduction to a family of transmission problems in r and by contraction arguments a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed an approximation for the associated vector field is also provided this shows that the motion on those manifolds is exponentially slow as in the well known case of the bistable equation however in opposition to the bistable case some of these manifolds are far from the attractor since these manifolds correspond to metastable patterns this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion it is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds

transport upscaling in heterogeneous aquifers what physical parameters control memory functions 1 power law tailing is often observed in the breakthrough curves btcs of tracer tests tailing is attributed to heterogeneity of aquifer properties and cannot be properly modeled by means of the homogeneous advection dispersion equation mass transfer models e g continuous time random walk method multirate mass transfer or fractional order advection dispersion equations using memory have been widely applied for reproducing observed tails the relationship between memory parameters obtained from btc fitting and the parameters characterizing the heterogeneity of hydraulic properties is still unclear here we investigate the conditions under which heterogeneity produces the type of tailing observed in the field and how memory functions are influenced by measurable heterogeneity parameters e g variance variogram or integral scale of the underlying transmissivity field we find that the slope of a btc in a log log plot is mainly influenced by the connectivity of the underlying permeability field but is insensitive to its variance the slope btc reaches asymptotically 2 as connectivity increases we conclude that an appropriate choice of the memory function allows reproducing the spreading caused by hydraulic heterogeneity but not necessarily the rate of mixing

delineation of connectivity structures in 2 d heterogeneous hydraulic conductivity fields connectivity is a critical aquifer property controlling anomalous transport behavior at large scales but connectivity cannot be easily defined in a continuous field based on information of the hydraulic conductivity alone we conceptualize it as a connecting structure a connected subset of a continuous hydraulic conductivity field that consists of paths of least hydraulic resistance we develop a simple and robust numerical method to delineate the connectivity structure using information of the hydraulic conductivity field only first the topology of the connectivity structure is determined by finding the path s of least resistance between two opposite boundaries and second a series of connectivity structures are created by inflating and shrinking the individual channels finally we apply this methodology to different heterogeneous fields we show that our method captures the main flow channels as well as the pathways of early time solute arrivals we find our method informative to study connectivity in 2 d heterogeneous hydraulic conductivity fields

local shannon entropy measure with statistical tests for image randomness in this paper we propose a new image randomness measure using shannon entropy over local image blocks the proposed local shannon entropy measure overcomes several weaknesses of the conventional global shannon entropy measure including unfair randomness comparisons between images of different sizes failure to discern image randomness before and after image shuffling and possible inaccurate scores for synthesized images statistical tests pertinent to this new measure are also derived this new measure is therefore both quantitative and qualitative the parameters in the local shannon entropy measure are further optimized for a better capture of local image randomness the estimated statistics and observed distribution from 50 000 experiments match the theoretical ones finally two examples are given applying the proposed measure to image randomness among shuffled images and encrypted images both examples show that the proposed method is more effective and more accurate than the global shannon entropy measure

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

minimum relative entropy forward probabilistic modeling the pioneering work of jaynes in bayesian maximum entropy methods has been successfully explored in many disciplines the principle of maximum entropy pme is a powerful and versatile tool of inferring a probability distribution from constraints that do not completely characterize that distribution minimum relative entropy mre is a method which has all the important attributes of the maximum entropy approach with the advantage that prior information may be easily included in this paper we use mre to determine the prior probability density function pdf of a set of model parameters based on limited information the resulting pdf is used in monte carlo simulations to provide expected values in field variables such as concentration and confidence limits we compare the probabilistic results from a traditional advection dispersion ade model based on volumetric averaging concepts with that of a model based on the assumption that the hydraulic conductivity is a stationary stochastic process the results suggest that although naff s 1990 model satisfies the observed data to a better degree than ade model the upper and lower confidence bands about the mean value are larger than the ade results this result we attribute to the fact that naff s 1990 model simply contains more parameters each of which is unknown and has to be estimated there is no statistical difference between the expected values of second spatial moments for the two models the examples presented in this paper illustrate problems associated with assigning gaussian pdfs as priors in a probabilistic model first such an assumption for the input parameters actually injects more information into the problem than may actually exist whether consciously or unconsciously this fact is borne out by comparison with minimum relative entropy theory second the output parameters as suggested from the monte carlo analysis cannot be assumed to be gaussian distributed even when the prior pdf is gaussian in form in a practical setting the significance of this result and the approximation of gaussian form would depend on the toxicity and environmental standards that apply to the problem

on the local and global comparison of generalized bajraktarevic means abstract given two continuous functions f g i r such that g is positive and f g is strictly monotone a measurable space t a a measurable family of d variable means m i d t i and a probability measure on the measurable sets a the d variable mean m f g m i d i is defined by m f g m x f g 1 t f m x 1 x d t d t t g m x 1 x d t d t x x 1 x d i d the aim of this paper is to study the local and global comparison problem of these means i e to find conditions for the generating functions f g and h k for the families of means m and n and for the measures such that the comparison inequality m f g m x m h k n x x i d be satisfied

slow motion in higher order systems and g convergence in one space dimension the slow motion in higher order systems and convergence in one space dimension was studied some of the theorems and computations that support and proves the stability of the equations are presented

asymptotics for a parabolic double obstacle problem we consider a parabolic double obstacle problem which is a version of the allen cahn equation u t delta u epsilon 2 psi prime u in omega times 0 infty where omega is a bounded domain epsilon is a small constant and psi is a double well potential here we take psi such that psi u 1 u 2 when u leq 1 and psi u infty when u 1 we study the asymptotic behaviour as epsilon rightarrow 0 of the solution of the double obstacle problem under some natural restrictions on the initial data we show that after a short time of order epsilon 2 ln epsilon the solution takes value 1 in a region omega t and value 1 in omega t where the region omega backslash omega t cup omega t is a thin strip and is contained in either a o epsilon ln epsilon or o epsilon neighbourhood of a hypersurface gamma t which moves with normal velocity equal to its mean curvature we also study the asymptotic behaviour as t rightarrow infty of the solution in the one dimensional case in particular we prove that the omega limit set consists of a singleton

bifurcations and multiple traffic jams in a car following model with reaction time delay we investigate an optimal velocity car following model for n cars on a circular singlelane road where reaction time delay of drivers is taken into account the stability of the uniform ow equilibrium is studied analytically while bifurcating periodic solutions for difierent wave numbers are investigated with numerical continuation techniques this reveals that the periodic solution with the smallest wave number may be stable and all other periodic solutions are unstable as n is increased periodic solutions develop stop and go fronts that correspond to rapid deceleration and acceleration between regions of uniformly owing and stagnant tra c in terms of the positions of all cars on the ring these fronts are associated with tra c jams all tra c jams form a tra c pattern that evolves under time due to slow motion of the fronts the tra c pattern corresponding to the stable periodic motion of cars is the only stable one however we flnd that other periodic orbits may be unstable only so weakly that they give rise to transient tra c jams that may persist for long times eventually such tra c jams either merge with one another or disperse until the stable tra c pattern is reached

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

metastable solutions for the thin interface limit of a phase field model abstract we consider a generalization of the semilinear phase field model from jimenez casas and rodriguez bernal proc roy soc edinburgh 129a 1999 571 600 for a more general enthalphy function we prove the existence of metastable solutions that evolve very slowly in time

differences in the scale dependence of dispersivity and retardation factors estimated from forced gradient and uniform flow tracer tests in three dimensional physically and chemically heterogeneous porous media 1 parameters estimated from forced gradient tracer tests are often used to simulate the transport of large contaminant plumes under natural gradients yet the difference in the flow system and the plume size is important recent intermediate scale experiments and numerical simulations in two dimensions have suggested that dispersivities from forced gradient tracer tests involving a small source may not be adequate to simulate the migration of a large plume under natural gradient conditions the differences in the scale dependence of effective dispersivities aeff for conservative and linearly sorbing solutes and retardation factors reff due to changes in the flow configuration and source size are investigated here on the basis of numerical simulations in three dimensional physically and chemically heterogeneous porous media it is shown that discrepancies observed in two dimensions due to small source size and nonuniform flow conditions are not so dramatic in typical field settings because a small vertical correlation scale permits a tracer source to sample the medium heterogeneity more effectively we further analyzed the benefit of using multiple injections in tracer tests by evaluating 1 the reduction in dispersivity due to the small source size and 2 the uncertainty in dispersivity estimates coefficient of variation the distance scales at which aeff and reff from single and multiple injection convergent flow tracer tests approach constant values are significantly greater than analogous values for natural gradients when dispersivities are estimated from convergent flow tracer tests involving multiple injections asymptotic aeff values are the same as those obtained from large source natural gradient tests in both convergent flows and natural gradient systems in the case of a negative correlation between lnk and lnkd random fields reff estimated from temporal moments of breakthrough curves decrease with distance while reff estimated from the center of mass velocities of the tracer plumes increase with time both asymptotically approaching the arithmetic mean of the retardation factor

inequalities for sums of powers be valid for all n e n and x x 0 it is necessary ana s cient thut min a b min c d and max a 6 d max c d 5 in i brenner showed independently that 5 is a sufficient condition for 4 he also obtained a number of more general results there exist other generalizations of theorem a see for instance losonczi 3 pales s but now we show one way that can also be applied to the inequality 1 assume that a b and c d and then rearrange 4 to obtain l q xy h xf p h x xt xi i xf x i cd c it seems to be natural to consider the following more general inequality

computer aided solution of the invariance equation for two variable gini means our aim is to solve the so called invariance equation in the class of two variable gini means g p q p email protected r i e to find necessary and sufficient conditions on the 6 parameters a b c d p q such that the identity g p q g a b x y g c d x y g p q x y x email protected r be valid we recall that for p q the gini mean g p q is defined by g p q x y x p y px q y q 1 p q x email protected r the proof uses the computer algebra system maple v release 9 to compute a taylor expansion up to 12th order which enables us to describe all the cases of the equality

generation and propagation of interfaces in reaction diffusion systems this paper is concerned with the asymptotic behavior as e0 of the solution u e e of the second initial boundary value problem of the reaction diffusion system u t e e u e 1 e f u e e 1 e u e 1 u e2 e t e e u e e where 0 is a constant when 2 3 9 2 3 9 f is bistable in the sense that the ordinary differential equation u t f u has two stable solutions u h and u h and one unstable solution u h 0 where h h 0 and h are three solutions of the algebraic equation f u 0

realization of meander permutations by boundary value problems abstract we consider neumann boundary value problems of the form uxx f x u ux 0 on the unit interval 0 x 1 for a certain class of dissipative nonlinearities f associated to these problems we have i meanders in the phase space u ux r 2 which are connected oriented simple curves on the plane intersecting a fixed oriented line the u axis in n points corresponding to the solutions and ii meander permutations f s n obtained by ordering the intersection points first along the u axis and then along the meander the meander permutation f is the permutation defined by the braid of solutions in the space x u ux it was recently shown by fiedler and rocha that f determines the global attractor of the dynamical system generated by the semilinear parabolic differential equation ut uxx f x u ux up to c0 orbit equivalence therefore these permutations are of considerable importance in the classification problem of the morse smale attractors for these dynamical systems in this paper we present a purely combinatorial characterization of the set of meander permutations that are realizable by the above boundary value problems

a mean field equation of motion for the dynamic ising model a mean field type of approximation is used to derive two differential equations one approximately representing the average behavior of the ising model with glauber spin flip stochastic dynamics and the other doing the same for kawasaki spin exchange dynamics the proposed new equations are compared with the cahn allen and cahn hilliard equations representing the same systems and with information about the exact behavior of the microscopic models

evaporation of droplets in the two dimensional ginzburg landau equation abstract we consider the problem of coarsening in two dimensions for the real scalar ginzburg landau equation this equation has exactly two stable stationary solutions the constant functions 1 and 1 we assume most of the initial condition is in the 1 phase with islands of 1 phase we use invariant manifold techniques to prove that the boundary of a circular island moves according to allen cahn curvature motion law we give a criterion for non interaction of two arbitrary interfaces and a criterion for merging of two nearby interfaces

strong monotonicity for various means abstract point wise monotonicity in parameters for various one parameter families of scalar means such as power difference means binomial means and stolarsky means is well known but norm comparison for corresponding operator means requires monotonicity in the sense of positive definiteness among other things we obtain monotonicity in the sense of infinite divisibility which is much stronger than that in the sense of positive definiteness these strong monotonicity results are proved based on explicit computations for measures in relevant levy khintchine or actually kolmogorov formulas

explicit construction of an inertial manifold for a reaction diffusion equation abstract an inertial manifold is constructed for the scalar reaction diffusion equation u t vu xx u with a cubic nonlinearity uniform bounds are obtained for the number of zeros along solutions to the variational equations satisfied by the difference of two elements on the unstable manifolds of equilibria this uniformity leads to the global parameterization of the attractor as a function defined in the linear unstable manifold of the least stable equilibrium by the introduction of local techniques near each equilibrium we succeed in constructing an inertial manifold of lowest possible dimension

almost global existence and transient self similar decay for poiseuille flow at criticality for exponentially long times abstract we consider nonlinear diffusion equations with critical exponent as tu x2u u3 with x r for small initial data in l1 l it is well known that almost all solutions of this system explode in finite time however we make the observation that in terms of the norm of the initial conditions it takes an exponentially long time moreover before explosion the l norm of such solutions becomes exponentially small which makes it almost impossible to observe the instability in experiments as an application we consider the long time transient self similar decay to unstable poiseuille flow at criticality for exponentially long times this together with a subcritical bifurcation and short time transient amplification is a principal obstruction in all attempts to measure the critical reynolds number for this experiment more and more precisely

an indirect assessment on the impact of connectivity of conductivity classes upon longitudinal asymptotic macrodispersivity solute transport takes place in heterogeneous porous formations with the log conductivity y ln k modeled as a stationary random space function of given univariate normal probability density fu

inequalities for hyperbolic functions abstract several inequalities involving hyperbolic functions are derived some of them are obtained with the aid of stolarsky and gini means

local monotonicity properties of two variable gini means and the comparison theorem revisited abstract in this paper we examine local monotonicity properties of the so called gini means defined by g a b x y x a y a x b y b 1 a b if a b exp x a log x y a log y x a y a if a b with respect to the parameters a b as a consequence of our results we obtain a new proof of the comparison theorem of two variable gini means

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

geological entropy and solute transport in heterogeneous porous media we propose a novel approach to link solute transport behavior to the physical heterogeneity of the aquifer which we fully characterize with two measurable parameters the variance of the log k values math formula and a new indicator hr that integrates multiple properties of the k field into a global measure of spatial disorder or geological entropy from the results of a detailed numerical experiment considering solute transport in k fields representing realistic distributions of hydrofacies in alluvial aquifers we identify empirical relationship between the two parameters and the first three central moments of the distributions of arrival times of solute particles at a selected control plane the analysis of experimental data indicates that the mean and the variance of the solutes arrival times tend to increase with spatial disorder i e hr increasing while highly skewed distributions are observed in more orderly structures i e hr decreasing or at higher math formula we found that simple closed form empirical expressions of the bivariate dependency of skewness on hr and math formula can be used to predict the emergence of non fickian transport in k fields considering a range of structures and heterogeneity levels some of which based on documented real aquifers the accuracy of these predictions and in general the results from this study indicate that a description of the global variability and structure of the k field in terms of variance and geological entropy offers a valid and broadly applicable approach for the interpretation and prediction of transport in heterogeneous porous media

on a reaction diffusion equation with robin and free boundary conditions this paper studies the following problem ut uxx f u 0 0 u 0 t bux 0 t t 0 u h t t 0 h t ux h t t t 0 h 0 h0 u x 0 x 0 x h0 where f is an unbalanced bistable nonlinearity b 0 0 and is a compactly supported c2 function we prove that there exists 0 such that vanishing happens when 0 and u t converges as t to 0 uniformly in 0 h t spreading happens when i e h t c t tends to a constant for some c 0 u t converges to a positive stationary solution locally uniformly in 0 and to a traveling semi wave with speed c near x h t in the transition case when u t v t h2 0 h t tends to 0 as t where t is a maximum point of u t and v is the unique even positive solution of v f v 0 subject to v 0 moreover with respect to b and f t pln t q o 1 for some p 0 and q r or t z for some root z of v z bv z

significance of higher moments for complete characterization of the travel time probability density function in heterogeneous porous media using the maximum entropy principle 1 the travel time formulation of advective transport in heterogeneous porous media is of interest both conceptually e g for incorporating retention processes and in applications where typically the travel time peak early and late arrivals of contaminants are of major concern in a regulatory or remediation context furthermore the travel time moments are of interest for quantifying uncertainty in advective transport of tracers released from point sources in heterogeneous aquifers in view of this interest the travel time distribution has been studied in the literature however the link to the hydraulic conductivity statistics has been typically restricted to the first two moments here we investigate the influence of higher travel time moments on the travel time probability density function pdf in heterogeneous porous media combining monte carlo simulations with the maximum entropy principle the monte carlo experimental pdf is obtained by the adaptive fup monte carlo method afmcm for advective transport characterized by a multi gaussian structure with exponential covariance considering two injection modes in flux and resident and lnk variance up to 8 a maximum entropy maxent algorithm based on fup basis functions is used for the complete characterization of the travel time pdf all travel time moments become linear with distance initial nonlinearity is found mainly for the resident injection mode which exhibits a strong nonlinearity within first 5iy for high heterogeneity for the resident injection mode the form of variance and all higher moments changes from the familiar concave form predicted by the first order theory to a convex form for the in flux mode linearity is preserved even for high heterogeneity the number of moments sufficient for a complete characterization of the travel time pdf mainly depends on the heterogeneity level mean and variance completely describe travel time pdf for low and mild heterogeneity skewness is dominant for lnk variance around 4 while kurtosis and fifth moment are required for lnk variance higher than 4 including skewness seems sufficient for describing the peak and late arrivals linearity of travel time moments enables the prediction of asymptotic behavior of the travel time pdf which in the limit converges to a symmetric distribution and fickian transport however higher order travel time moments may be important for most practical purposes and in particular for advective transport in highly heterogeneous porous media for a long distance from the source

comparison of means generated by two functions and a measure abstract given two continuous functions f g i r such that g is positive and f g is strictly monotone and a probability measure on the borel subsets of 0 1 the two variable mean m f g i 2 i is defined by m f g x y f g 1 0 1 f t x 1 t y d t 0 1 g t x 1 t y d t x y i the aim of this paper is to study the comparison problem of these means i e to find conditions for the generating functions f g and h k and for the measures such that the comparison inequality m f g x y m h k x y x y i holds

invariant manifolds for metastable patterns in ut e2uxx f u we consider the above equation on the interval 0 x 1 subject to neumann boundary conditions with f u f u where f is a double well energy density function with equal minima our previous work 3 proved the existence and persistence of very slowly evolving patterns metastable states in solutions with two phase initial data here we characterise these metastable states in terms of the global unstable manifolds of equilibria as conjectured by fusco and hale 6

origins of anomalous transport in heterogeneous media structural and dynamic controls anomalous or non fickian transport is ubiquitous in the context of tracer migration in geological formations we quantitatively identify the origin of anomalous transport in a representative model of a heterogeneous porous medium under uniform in the mean flow conditions we focus on anomalous transport which arises in the complex flow patterns of lognormally distributed hydraulic conductivity k fields with several decades of k values transport in the domains is determined by a particle tracking technique and characterized by breakthrough curves btcs the btc averaged over multiple realizations demonstrates anomalous transport in all cases which is accounted for entirely by a power law distribution t 1 of local transition times the latter is contained in the probability density function t of transition times embedded in the framework of a continuous time random walk ctrw a unique feature of our analysis is the derivation of t as a function of parameters quantifying the heterogeneity of the domain in this context we first establish the dominance of preferential pathways across each domain and characterize the statistics of these pathways by forming a particle visitation weighted histogram hw k of the hydraulic conductivity by converting the ln k dependence of hw k into time we demonstrate the equivalence of hw k and t and delineate the region of hw k that forms the power law of t this thus defines the origin of anomalous transport analysis of the preferential pathways clearly demonstrates the limitations of critical path analysis and percolation theory as a basis for determining the origin of anomalous transport furthermore we derive an expression defining the power law exponent in terms of the hw k parameters the equivalence between hw k and t is a remarkable result particularly given the nature of the k heterogeneity the complexity of the flow field within each realization and the statistics of the particle transitions

self similarity in a coarsening model in one dimension motivated from asymptotic laws of motion for transition layers in the equation u t 2 u xx u u 2 we consider the following model for coarsening of a fine partition of an interval find the shortest subinterval of the partition and joint it with its neighbours combining three into one making a random order assumption we develop and study an unusual coagulation equation for the distribution of interval lengths we establish the existence of a self similar solution of this equation by using laplace transform techniques simulation data indicate that random order does persist if present initially and the distribution approaches similarity form

solutions to the nonautonomous bistable equation with specified morse index part i existence we investigate the existence of unstable solutions of specified morse index for the equation e 2 u xx f x u 0 on a finite interval and neumann boundary conditions

slow motion for gradient systems with equal depth multiple well potentials abstract for scalar reaction diffusion in one space dimension it has been known for a long time that fronts move with an exponentially small speed for potentials with several distinct minimizers the purpose of this paper is to provide a similar result in the case of systems our method relies on a careful study of the evolution of localized energy this approach also has the advantage of relaxing the preparedness assumptions on the initial datum

inequalities for differences of powers on etudie l inegalite 1 x a 1 y a 1 a 1 1 x a k y a k a k k avec 0 1 k on deduit des conditions necessaires de validite pour tout x y 0 avec x y on demontre que ces conditions necessaires sont aussi suffisantes

schur convexity and schur geometrically concavity of gini means the schur convexity and the schur geometric convexity with variables x y r 2 for fixed s t of gini means g r s x y are discussed some new inequalities are obtained

finite size scaling analysis of percolation in three dimensional correlated binary markov chain random fields percolation and finite size scaling properties in three dimensional binary correlated markov chain random fields on a cubic lattice are computed by extensive monte carlo simulation at short correlation scales the percolation threshold in correlated random fields decreases as the correlation scale increases the rate of decrease rapidly diminishes for correlation lengths larger than 2 3 lattice sites at correlation scales of 4 6 lattice sites the percolation threshold is found to be 0 126 0 001 for the markov chain random fields similar to that for sequential gaussian and indicator random fields which are evaluated for comparison the average percolation threshold in finite size lattices is a function of both the correlation length and the finite lattice size the universal scaling constants for mean cluster size and backbone fraction are found to be consistent with results on uncorrelated lattices but prefactors of scaling relationships vary with correlation length the squared radius of gyration of non percolating clusters is found to scale with gamma nu and its scaling prefactors are independent of the correlation scale prefactors are similar between the three random field generators evaluated the percolation properties derived here are useful to account for finite size effects on percolation in natural or man made correlated systems

hyperbolic periodic orbits in nongradient systems and small noise induced metastable transitions abstract small noise can induce rare transitions between metastable states which can be characterized by maximum likelihood paths mlps nongradient systems contrast gradient systems in that mlp does not have to cross the separatrix at a saddle point but instead possibly at a point on a hyperbolic periodic orbit a numerical approach for identifying such unstable periodic orbits is proposed based on string method in a special class of nongradient systems orthogonal type there are provably local mlps that cross such saddle point or hyperbolic periodic orbit and the separatrix crossing location determines the associated local maximum of transition rate in general cases however the separatrix crossing may not determine a unique local maximum of the rate as we numerically observed a counter example in a sheared 2d space allen cahn spde it is a reasonable conjecture that there are always local mlps associated with each attractor on the separatrix such as saddle point or hyperbolic periodic orbit our numerical experiments did not disprove so

reaction diffusion systems coupled at the boundary and the morse smale property abstract we study an one dimensional nonlinear reaction diffusion system coupled on the boundary such system comes from modeling problems of temperature distribution on two bars of same length jointed together with different diffusion coefficients we prove the transversality property of unstable and stable manifolds assuming all equilibrium points are hyperbolic to this end we write the system as an equation with noncontinuous diffusion coefficient we then study the nonincreasing property of the number of zeros of a linearized nonautonomous equation as well as the sturm liouville properties of the solutions of a linear elliptic problem