scaling of the distribution of shortest paths in percolation we present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for i the effect of the finite size of the system and ii the dependence of this distribution on the site occupancy probability p we test the hypothesis for the case of two dimensional percolation

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

statistical properties for flows with unbounded roof function including the lorenz attractor for geometric lorenz attractors including the classical lorenz attractor we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors we also obtain the functional central limit theorem and moment estimates as well as iterated versions of these results a consequence is deterministic homogenisation convergence to a stochastic differential equation for fast slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two

semismooth karush kuhn tucker equations and convergence analysis of newton and quasi newton methods for solving these equations there are several forms of systems of nonsmooth equations which are equivalent to the karush kuhn tucker kkt system of a nonlinearly constrained optimization problem nlp if the nlp is twice continuously differentiable and the hessian functions of its objective and constraint functions are locally lipschitzian then these kkt equations are strongly semismooth if furthermore the linear independence condition and the strong second order sufficiency condition are satisfied at a kkt point then the generalized jacobians of these kkt equations are nonsingular at that point and the sequence generated by the generalized newton method converges to this point q quadratically however direct application of quasi newton methods cannot guarantee q superlinear convergence we present a mixed quasi newton method which converges q superlinearly with common symmetrical updating rules under the above conditions for the generalized newton method superlinear convergence of the primal variables and global convergence are also discussed

method for generating long range correlations for large systems we propose a new method to generate a sequence of random numbers with long range power law correlations that overcomes known difficulties associated with large systems the new method presents an improvement on the commonly used methods we apply the algorithm to generate enhanced diffusion isotropic and anisotropic self affine surfaces and isotropic and anisotropic correlated percolation

the ergodicity of a hamiltonian system of two particles in an external field abstract for a hamiltonian system of two one dimensional balls in an external field we prove the ergodicity if the total energy of the system is fixed our proof is an extension of that applied earlier to dispersing billiard systems

how high dimensional stadia look like we give the affirmative answer to the long standing question whether or not the mechanism of defocusing can produce a chaotic behavior in high dimensional hamiltonian systems to do this we prove that billiards in a class of regions in r n n 2 with focusing and flat boundary components have nonvanishing lyapunov exponents

water propagation in two dimensional petroleum reservoirs abstract in the present paper we investigate the problem of water propagation in 2 dimensional 2d petroleum reservoir in which each site has the probability p of being occupied we first analyze this propagation pattern described by darcy equations by focusing on its geometrical features we find that the domain walls of this model at p p c 0 59 are schramm loewner evolution sle curves with 3 05 0 1 consistent with the ising universality class we also numerically show that the fractal dimension of these domain walls at p p c is d f 1 38 consistent with sle 3 along with this analysis we introduce a self organized critical soc model in which the water movement is modeled by a chain of topplings taking place when the water saturation exceeds the critical value we present strong indications that it coincides with the reservoir simulation described by darcy equation we further analyze the soc model and show numerically that for this model the spanning cluster probability has a maximum around p 0 65

ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2 torus we study generic piecewise linear hyperbolic automorphisms of the 2 torus we explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations

central limit theorem and recurrence for random walks in bistochastic random environments we prove the annealed central limit theorem for random walks in bistochastic random environments on zd with zero local drift the proof is based on a dynamicist s interpretation of the system and requires a much weaker condition than the customary uniform ellipticity moreover recurrence is derived for d 2

postbreakthrough behavior in flow through porous media we numerically simulate the traveling time of a tracer in convective flow between two points injection and extraction separated by a distance r in a model of porous media d 2 percolation we calculate and analyze the traveling time probability density function for two values of the fraction of connecting bonds p the homogeneous case p 1 and the inhomogeneous critical threshold case p p c we analyze both constant current and constant pressure conditions at p p c the homogeneous p 1 case serves as a comparison base for the more complicated p p c situation we find several regions in the probability density of the traveling times for the homogeneous case p 1 and also for the critical case p p c for both constant pressure and constant current conditions for constant pressure the first region i p corresponds to the short times before the flow breakthrough occurs when the probability distribution is strictly zero the second region ii p corresponds to numerous fast flow lines reaching the extraction point with the probability distribution reaching its maximum the third region iii p corresponds to intermediate times and is characterized by a power law decay the fourth region iv p corresponds to very long traveling times and is characterized by a different power law decaying tail the power law characterizing region iv p is related to the multifractal properties of flow in percolation and an expression for its dependence on the system size l is presented the constant current behavior is different from the constant pressure behavior and can be related analytically to the constant pressure case we present theoretical arguments for the values of the exponents characterizing each region and crossover times our results are summarized in two scaling assumptions for the traveling time probability density one for constant pressure and one for constant current we also present the production curve associated with the probability of traveling times which is of interest to oil recovery

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

semi focusing billiards hyperbolicity in this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of 3 we also prove that a related class of semi focusing billiards has mixed dynamics i e their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other these billiards are the first rigorous examples of billiards in domains of 3 with divided phase space

conditional proof of the boltzmann sinai ergodic hypothesis we consider the system of n 2 elastically colliding hard balls of masses m1 mn and radius r on the flat unit torus mathbb t nu 2 we prove the so called boltzmann sinai ergodic hypothesis i e the full hyperbolicity and ergodicity of such systems for every selection m1 mn r of the external parameters provided that almost every singular orbit is geometrically hyperbolic sufficient i e the so called chernov sinai ansatz is true the present proof does not use the formerly developed rather involved algebraic techniques instead it employs exclusively dynamical methods and tools from geometric analysis

design of hyperbolic billiards we formulate a general framework for the construction of hyperbolic billiards spherical symmetry is exploited for a simple treatment of billiards with spherical caps and soft billiards in higher dimensions other examples include the papenbrock stadium

an augmented lagrangian function with improved exactness properties in this paper we introduce a new exact augmented lagrangian function for the solution of general nonlinear programming problems for this lagrangian function a complete equivalence between its unconstrained minimization on an open set and the solution of the original constrained problem can be established under mild assumptions and without requiring the boundedness of the feasible set of the constrained problem moreover we describe an unconstrained algorithmic model which is globally convergent toward kkt pairs of the original constrained problem the algorithmic model can be endowed with a superlinear rate of convergence by a proper choice of the search direction in the unconstrained minimization without requiring strict complementarity

infinite horizon lorentz tubes and gases recurrence and ergodic properties we construct classes of two dimensional aperiodic lorentz systems that have infinite horizon and are chaotic in the sense that they are poincare recurrent uniformly hyperbolic ergodic and the first return map to any scatterer is k mixing in the case of the lorentz tubes i e lorentz gases in a strip we define general measured families of systems emph ensembles for which the above properties occur with probability 1 in the case of the lorentz gases in the plane we define families endowed with a natural metric within which the set of all chaotic dynamical systems is uncountable and dense

traveling length and minimal traveling time for flow through percolation networks with long range spatial correlations we study the distributions of traveling length l and minimal traveling time t mathrm min through two dimensional percolation porous media characterized by long range spatial correlations we model the dynamics of fluid displacement by the convective movement of tracer particles driven by a pressure difference between two fixed sites wells separated by euclidean distance r for strongly correlated pore networks at criticality we find that the probability distribution functions p l and p t mathrm min follow the same scaling ansatz originally proposed for the uncorrelated case but with quite different scaling exponents we relate these changes in dynamical behavior to the main morphological difference between correlated and uncorrelated clusters namely the compactness of their backbones our simulations reveal that the dynamical scaling exponents d l and d t for correlated geometries take values intermediate between the uncorrelated and homogeneous limiting cases where l ensuremath sim r d l and t mathrm min ensuremath sim r d t and l and t mathrm min are the most probable values of l and t mathrm min respectively

statistical behaviors for renormalization of correlated permeability field abstract in this article the statistical properties for the renormalized permeability obtained from the renormalization of the correlated permeability field are investigated in contrast to the uncorrelated porous media the scaling of the variance of the renormalized permeability field exhibits a crossover behavior when the correlation lengths are larger compared with the domain scale covered by the renormalization procedure the variance of the renormalized permeability will decrease slowly and the scaling exponent will be close to zero as the renormalization number increases the covered domain scale will eventually become larger than the correlation lengths and then the scaling property will transit to the uncorrelated case the convergent values of the renormalized permeability for isotropic and anisotropic correlated media are also investigated both the theoretical analysis and the simulation results show that larger correlation length in one direction will lead to a larger convergent value in the corresponding direction for the log normal permeability field numerical simulations show that the crossover scaling and also the convergent value for the renormalized permeability can be fitted very well by simple mathematical functions

traveling time and traveling length in critical percolation clusters we study traveling time and traveling length for tracer dispersion in two dimensional bond percolation modeling flow by tracer particles driven by a pressure difference between two points separated by euclidean distance r we find that the minimal traveling time t min scales as t min approximately r 1 33 which is different from the scaling of the most probable traveling time t approximately r 1 64 we also calculate the length of the path corresponding to the minimal traveling time and find l min approximately r 1 13 and that the most probable traveling length scales as l approximately r 1 21 we present the relevant distribution functions and scaling relations

recurrence and higher ergodic properties for quenched random lorentz tubes in dimension bigger than two we consider the billiard dynamics in a non compact set of d that is constructed as a bi infinite chain of translated copies of the same d dimensional polytope a random configuration of semi dispersing scatterers is placed in each copy the ensemble of dynamical systems thus defined one for each global realization of the scatterers is called quenched random lorentz tube under some fairly general conditions we prove that every system in the ensemble is hyperbolic and almost every system is recurrent ergodic and enjoys some higher chaotic properties

serial and parallel computation of karush kuhn tucker points via nonsmooth equations in this paper a robust iterative algorithm is presented for computing karush kuhn tucker kkt points of nonlinear programs this algorithm is a variation of the ne sqp method for solving the nonlinear complementarily problem it makes use of a merit function that combines the original objective function of the nonlinear program with a residual function of the kkt conditions formulated as a system of nonsmooth equations global convergence and a q quadratic rate of convergence of the algorithm are established under some standard constraint qualifications in nonlinear programming theory but without the strict complementarily condition parallel implementations of the algorithm are discussed

global convergence on an active set sqp for inequality constrained optimization sequential quadratic programming sqp has been one of the most important methods for solving nonlinearly constrained optimization problems in this paper we present and study an active set sqp algorithm for inequality constrained optimization the active set technique is introduced which results in the size reduction of quadratic programming qp subproblems the algorithm is proved to be globally convergent thus the results show that the global convergence of sqp is still guaranteed by deleting some redundant constraints

potentials on the two torus for which the hamiltonian flow is ergodic we consider the motion of a test particle in a compound central potential field on a two dimensional torus we discuss three different classes of potentials attracting repelling and mixed that lead to hamiltonian systems which have positive lyapunov exponent almost everywhere and are ergodic included among the mixed potentials are smooth potentials without singularities

flow between two sites on a percolation cluster we study the flow of fluid in porous media in dimensions d52 and 3 the medium is modeled by bond percolation on a lattice of l d sites while the flow front is modeled by tracer particles driven by a pressure difference between two fixed sites wells separated by euclidean distance r we investigate the distribution function of the shortest path connecting the two sites and propose a scaling ansatz that accounts for the dependence of this distribution i on the size of the system l and ii on the bond occupancy probability p w e confirm by extensive simulations that the ansatz holds for d52 and 3 further we study two dynamical quantities i the minimal traveling time of a tracer particle between the wells when the total flux is constant and ii the minimal traveling time when the pressure difference is constant a scaling ansatz for these dynamical quantities also includes the effect of finite system size l and off critical bond occupation probability p we find that the scaling form for the distribution functions for these dynamical quantities for d5 2a nd 3i s similar to that for the shortest path but with different critical exponents our results include estimates for all parameters that characterize the scaling form for the shortest path and the minimal traveling time in two and three dimensions these parameters are the fractal dimension the power law exponent and the constants and exponents that characterize the exponential cutoff functions