continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

exact critical exponent for the shortest path scaling function in percolation it is shown that the critical exponentg1 related to pair connectiveness and shortest path or chemical distance scaling recently studied by porto et al dokholyan et al and grassberger can be found exactly in two dimensions by using a crossing probability result of cardy with the outcomeg1d 25 24 this prediction is consistent with existing simulation results

scaling for the critical percolation backbone we study the backbone connecting two given sites of a two dimensional lattice separated by an arbitrary distance r in a system of size l at the percolation threshold we find a scaling form for the average backbone mass approximately l db g r l where g can be well approximated by a power law for 0 approximately l db psi r psi for the entire range 0 r l we also propose a scaling form for the probability distribution p m b of backbone mass for a given r for r approximately l p mb is peaked around l db whereas for r l p m b decreases as a power law m taub b with taub approximately 1 20 0 03 the exponents psi and taub satisfy the relation psi db taub 1 and psi is the codimension of the backbone psi d db

correlation and complexity analysis of well logs via lyapunov hurst lempel ziv and neural network algorithms abstract well logs produce a wealth of data that can be used to evaluate the production capacity of oil and gas fields these data are usually concerned with depth series of petrophysical quantities such as the sonic transient time gamma emission deep induction resistivity neutron porosity and bulk density here we perform a correlation and complexity analysis of well log data from the namorado s school field using lyapunov hurst lempel ziv and neural network algorithms after identifying the most correlated and complex series we demonstrate that well log data estimates can be confidently performed by neural network algorithms either to complete missing data or to infer complete well logs of a specific quantity

threshold dynamics for the piecewise constant mumford shah functional we propose an efficient algorithm for minimizing the piecewise constant mumford shah functional of image segmentation it is based on the threshold dynamics of merriman bence and osher for evolving an interface by its mean curvature we show that a very fast minimization can be achieved by alternating the solution of a linear parabolic partial differential equation and simple thresholding

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

the fractal dimension of the minimum path in two and three dimensional percolation the authors calculate the fractal dimension d min of the shortest path l between two points on a percolation cluster where l approximately rd min and r is the pythagorean distance between the points they find d min 1 130 or 0 002 for d 2 and 1 34 or 0 01 for d 3

fuzzy data envelopment analysis and its application to location problems in this paper fuzzy dea data envelopment analysis models are proposed for evaluating the efficiencies of objects with fuzzy input and output data the obtained efficiencies are also fuzzy numbers that reflect the inherent ambiguity in evaluation problems under uncertainty an aggregation model for integrating fuzzy attribute values is provided in order to rank objects objectively using the proposed method a case study involving a restaurant location problem is analyzed in detail rent of establishment traffic amount level of security consumer consumption level and competition level are identified as the primary factors in determining an ideal location for a japanese style rotisserie restaurant based on field investigation the uncertain information on primary factors is represented by fuzzy numbers using the fuzzy aggregation model the best location of restaurant is determined the case study shows that fuzzy dea models can be quite useful for solving business problems under uncertainty

an error estimate for a new scheme for mean curvature motion in this work we propose a new numerical scheme for the anisotropic mean curvature equation the solution of the scheme is not unique but for all numerical solutions we provide an error estimate between the continuous solution and the numerical approximation our scheme is also applicable to computing the solution to dislocation dynamics equations we also provide some numerical simulations

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

a convergent monotone difference scheme for motion of level sets by mean curvature an explicit convergent finite difference scheme for motion of level sets by mean curvature is presented the scheme is defined on a cartesian grid using neighbors arranged approximately in a circle the accuracy of the scheme which depends on the radius of the circle dx and on the angular resolution d is formally o dx2 d the scheme is explicit and nonlinear the update involves computing the median of the values at the neighboring grid points numerical results suggest that despite the low accuracy acceptable results are achieved for small stencil sizes a numerical example is presented which shows that the centered difference scheme is non convergent

multifractal properties of the random resistor network we study the multifractal spectrum of the current in the two dimensional random resistor network at the percolation threshold we consider two ways of applying the voltage difference i two parallel bars and ii two points our numerical results suggest that in the infinite system limit the probability distribution behaves for small i as p i 1 i where i is the current as a consequence the moments of i of order q qc50 do not exist and all currents of value below the most probable one have the fractal dimension of the backbone the backbone can thus be described in terms of only i blobs of fractal dimension db and ii high current carrying bonds of fractal dimension going from 1 n to db

conductivity exponent and backbone dimension in 2 d percolation abstract we present high statistics simulations for 2 d percolation clusters in the bus bar geometry at the critical point for site and for bond percolation we measured their backbone sizes and electrical conductivities for all sets of measurements we find large corrections to scaling most of which do not seem to be described by single powers using single power terms for the corrections to scaling of the backbone masses we would obtain fractal dimensions which are different for site and bond percolation while the correct result is d b 1 6432 0 0008 for both for the conductivity the corrections to scaling are strongly non monotonic for bond percolation the exponent t t is measured as 0 9826 0 0008 in disagreement with the alexander orbach and other conjectures

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a variational method for multiphase volume preserving interface motions we develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases whose volumes are preserved throughout time the foundation of the method is a thresholding algorithm of the bence merriman osher type the original algorithm is reformulated in a vector setting which allows for a natural inclusion of constraints even in the multiphase case moreover a new method for overcoming the inaccuracy of thresholding methods on non adaptive grids is designed since this inaccuracy becomes especially prominent in volume preserving motions formal analysis of the method and numerical tests are presented

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

analysis of diffusion generated motion for mean curvature flow in codimension two a gradient flow approach the merriman bence osher mbo scheme also known as diffusion generated motion or thresholding is an efficient numerical algorithm for computing mean curvature flow mcf it is fairly well understood in the case of hypersurfaces this paper establishes the first convergence proof of the scheme in codimension two we concentrate on the case of the curvature motion of a filament curve in mathbb r 3 our proof is based on a new generalization of the minimizing movements interpretation for hypersurfaces esedoglu otto 15 by means of an energy that approximates the dirichlet energy of the state function as long as a smooth mcf exists we establish uniform energy estimates for the approximations away from the smooth solution and prove convergence towards this mcf the current result that holds in codimension two relies in a very crucial manner on a new sharp monotonicity formula for the thresholding energy this is an improvement of an earlier approximate version

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

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

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

a simple proof of convergence for an approximation scheme for computing motions by mean curvature we prove the convergence of an approximation scheme recently proposed by bence merriman and osher for computing motions of hypersurfaces by mean curvature our proof is based on viscosity solutions methods

distributions and moments of structural properties for percolation clusters we study numerically and by scaling methods the distributions and moments of several structural properties of percolation clusters in two and three dimensions the clusters are generated at criticality and properties such as the distribution of the mass as a function of linear size or chemical distance are studied our results suggest that the hierarchy of moments can be represented by a single gap exponent using a scaling approach we obtain analytical forms for the different distribution functions which agree very well with the numerical data

a convolution thresholding approximation of generalized curvature flows we construct a convolution thresholding approximation scheme for the geometric surface evolution in the case when the velocity of the surface at each point is a given function of the mean curvature conditions for the monotonicity of the scheme are found and the convergence of the approximations to the corresponding viscosity solution is proved we also discuss some aspects of the numerical implementation of such schemes and present several numerical results

spreading and backbone dimensions of 2d percolation the author presents results of high statistics simulations of the spreading of 2d percolation and of backbones of 2d percolation clusters while the algorithm employed for the spreading is more or less standard for identifying backbones the author uses a new and very efficient recursive algorithm the author uses new critical exponents to test previous conjectures

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

position space renormalisation group study of shortest paths in percolation clusters the average length of the shortest paths along the backbone of the incipient infinite cluster is expected to diverge near the percolation threshold pc as mod p pc mod nu s with nu s different from the correlation length exponent nu the author applies the position space renormalisation group psrg technique to obtain nu s both for square and cubic lattices and for a triangular lattice in two dimensions the results obtained nu s 1 49 for d 2 and 1 21 for d 3 are in very good agreement with the existing results obtained by different methods

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

convergence of a generalized fast marching method for an eikonal equation with a velocity changing sign we present a new fast marching algorithm for an eikonal equation with a velocity changing sign this first order equation models a front propagation in the normal direction the algorithm is an extension of the fast marching method in two respects the first is that the new scheme can deal with a time dependent velocity and the second is that there is no restriction on its change in sign we analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions finally we present some numerical simulations of fronts propagating in mathbb r 2

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

convergence of approximation schemes for nonlocal front propagation equations we provide a convergence result for numerical schemes approximating nonlocal front propagation equations our schemes are based on a recently investigated notion of a weak solution for these equations we also give examples of such schemes for a dislocation dynamics equation and for a fitzhugh nagumo type system

a diffusion generated approach to multiphase motion in this article we present a diffusion generated approach for evolving multiple junctions this work generalizes an earlier method by merriman bence and osher which alternately diffuses and sharpens characteristic functions for each phase region to produce pure mean curvature flow specifically our new method produces a normal velocity which depends on a positive multiple of the curvature of the interface plus the difference in bulk energy densities for prescribed junction angles this simple method naturally treats topological mergings and breakings produces no overlapping regions or vacuums and can be made very fast numerical studies are provided which show that our method agrees with front tracking and a recent variational approach for a variety of examples asymptotic expansions are also carried out near junctions to justify our algorithms

multiclass data segmentation using diffuse interface methods on graphs we present two graph based algorithms for multiclass segmentation of high dimensional data on graphs the algorithms use a diffuse interface model based on the ginzburg landau functional related to total variation and graph cuts a multiclass extension is introduced using the gibbs simplex with the functional s double well potential modified to handle the multiclass case the first algorithm minimizes the functional using a convex splitting numerical scheme the second algorithm uses a graph adaptation of the classical numerical merriman bence osher mbo scheme which alternates between diffusion and thresholding we demonstrate the performance of both algorithms experimentally on synthetic data image labeling and several benchmark data sets such as mnist coil and webkb we also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph laplacian and take advantage of the sparsity of the matrix experiments indicate that the results are competitive with or better than the current state of the art in multiclass graph based segmentation algorithms for high dimensional data