continuum percolation with discs having a distribution of radii it is shown that for continuum percolation with overlapping discs having a distribution of radii the net areal density of discs at percolation threshold depends non trivially on the distribution and is not bounded by any finite constant results of a monte carlo simulation supporting the argument are presented

continuum percolation threshold for interpenetrating squares and cubes monte carlo simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions simulations are performed for two cases i objects whose edges are aligned parallel to one another and ii randomly oriented objects for squares whose edges are aligned the critical area fraction at the percolation threshold ensuremath varphi c 0 6666 ifmmode pm else textpm fi 0 0004 while for randomly oriented squares ensuremath varphi c 0 6254 ifmmode pm else textpm fi 0 0002 6 smaller for cubes whose edges are aligned the critical volume fraction at the percolation threshold ensuremath varphi c 0 2773 ifmmode pm else textpm fi 0 0002 while for randomly oriented cubes ensuremath varphi c 0 2168 ifmmode pm else textpm fi 0 0002 22 smaller

the equivalence of the dla and a hydrodynamic model the authors prove that the growth process governed by the diffusion limited aggregation dla algorithm as well as by some of its natural extensions is isomorphous to the process of pushing an incompressible newtonian fluid by an inviscid fluid through a porous medium in the chamber tube ct model provided that the parameters of the ct model are properly chosen in this way they give a direct quantitative relation between hydrodynamics in random porous media and the fractal theory of stochastic dla like aggregates on the other hand this work presents an example of two equivalent dynamical systems one of them with stochastic dynamics applied to a deterministic medium and the other with deterministic dynamics applied to a random medium

a unified treatment of integer parameter models abstract it is shown that there is an analog to the score function the derivative of the log likelihood with respect to the parameter here called the difference score which plays an important theoretical role in integer parameter models a unified treatment of integer parameter models can be obtained by recognizing that many commonly used models have a linear difference score lds it is established that many of the exact results available for exponential family models also hold for lds models in particular we find an information lower bound show the uniqueness of the maximum likelihood estimator establish monotone likelihood ratio and establish optimality properties in two sided tests it is also shown that if one has a multiparameter likelihood in which there is a conditional likelihood depending on n alone then these exact results apply there as well it is also established when and why the usual likelihood methodology will apply to a class of nonstandard models in which the natural approxima

a note on n estimators for the binomial distribution abstract consider k success counts from a binomial distribution with unknown n and success probability p we examine the problem of estimating n by integrating the likelihood for n and p over a beta density for p we obtain the beta binomial distribution resulting in stable and reasonably efficient estimators of n which compare favorably with and are often better than the estimates introduced by olkin et al 1981

applications of fractal concepts in petroleum engineering abstract the different models of fluid transport in porous media are summarized and their applications in oil reservoir simulation are analysed it is shown that the problems for application of these models and of fractal concepts are of three types no validity of the models due to viscous forces always present in both fluids at large scale difficulty to use fractal geometry as a model of reservoir and finally the need for analytical or differential equations

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

fluid flow in porous media with rough pore solid interface quantifying fluid flow through porous media hinges on the description of permeability a property of considerable importance in many fields ranging from oil and gas exploration to hydrology a common building block for modeling porous media permeability is consideration of fluid flow through tubes with circular cross section described by poiseuille s law in which flow discharge is proportional to the fourth power of the tube s radius in most natural porous media pores are neither cylindrical nor smooth they often have an irregular cross section and rough surfaces this study presents a theoretical scaling of poiseuille s approximation for flow in pores with irregular rough cross section quantified by a surface fractal dimension ds2 the flow rate is a function of the average pore radius to the power 2 3 ds2 instead of 4 in the original poiseuille s law values of ds2 range from 1 to 2 hence the power in the modified poiseuille s approximation varies between 4 and 2 indicating that flow rate decreases as pore surface roughness and surface fractal dimension ds2 increases we also proposed pore length radius relations for isotropic and anisotropic fractal porous media the new theoretical derivations are compared with standard approximations and with experimental values of relative permeability the new approach results in substantially improved prediction of relative permeability of natural porous media relative to the original poiseuille equation

a critical exponent for shortest path scaling in continuum percolation we carry out monte carlo experiments to study the scaling behavior of shortest path lengths in continuum percolation these studies suggest that the critical exponent governing this scaling is the same for both continuum and lattice percolation we use splitting a technique that has not yet been fully exploited in the physics literature to increase the speed of our simulations this technique can also be applied to other models where clusters are grown sequentially

bayesian analyses for a multiple capture recapture model summary in multiple capture recapture surveys the probability of capture can vary between sampling occasions the model accounting for this variation is known as at bayes empirical bayes and bayes empirical bayes solutions are given to the problems of interval estimation decision making and point estimation of the population size n when the number of sampling occasions is small to moderate and the number of recaptured units observed on each sampling occasion is moderate estimates obtained from empirical bayes and bayes empirical bayes methods compare closely to bayesian methods using a reference prior distribution for the capture probabilities however when the number of sampling occasions is large and the number of recaptured units observed on each sampling occasion is small inferences obtained using different reference priors can differ considerably

inference for exponential order statistic models based on an integrated likelihood function abstract methods of statistical inference are developed for the exponential order statistic eos model where only a subset of order statistics from a collection of n iid exponential random detection times is observable when the rate parameter for detections is unknown the maximum likelihood estimator mle of the unknown integer parameter n can be infinite with substantial probability inference for n is developed using a pseudolikelihood function obtained by integrating out the rate parameter the estimator that maximizes this function called the integrated likelihood estimator ile is shown to be finite and to have better sampling properties than the mle parameter based asymptotics are developed for the case where n is large application of the methodology is illustrated using two datasets

an explanation of scale dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry stochastic transport theories rely on the assumptions that the principles of stationarity and ergodicity are satisfied these assumptions imply that the pattern of heterogeneity can be viewed as being spatially periodic thus yielding a finite correlation scale we examine a type of heterogeneity which does not satisfy these principles self similar or fractal heterogeneity which exhibits a pattern over a large range of scales that is independent of the scale of observation and thus will possess a very large correlation scale we review the basic concepts of fractal geometry and develop scaling relationships for fractal travel distance versus scale of observation we then develop lagrangian models for dispersion in a single fractal streamtube and for a set or bundle of fractal streamtubes the results are compared to classical one dimensional advection dispersion theory existing fractal and stochastic theories and to a summary of field measured dispersivity data for the single streamtube model field measured dispersivity is proportional to the straight line travel distance raised to a power of d 1 where d is the fractal dimension for the set of fractal streamtubes field measured dispersivity is proportional to the straight line travel distance raised to a power of 2d 1 the analytical expressions are verified and elaborated by a fractal random walk computer model the computer model is used to show the self similar nature of heterogeneity that is independent of the scale of observation comparisons of the field dispersivity data with our fractal models indicate that most tracer tests have been performed in media that approximate a set of fractal streamtubes

continuum percolation thresholds in two dimensions a wide variety of methods have been used to compute percolation thresholds in lattice percolation the most powerful of these methods consists of microcanonical simulations using the union find algorithm to efficiently determine the connected clusters and in two dimensions using exact values from conformal field theory for the probability at the phase transition that various kinds of wrapping clusters exist on the torus we apply this approach to percolation in continuum models finding overlaps between objects with real valued positions and orientations in particular we find precise values of the percolation transition for disks squares rotated squares and rotated sticks in two dimensions and confirm that these transitions behave as conformal field theory predicts the running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality

modeling of scale dependent dispersion in hydrogeologic systems an approach for handling scale dependent dispersion using a finite element linear triangular solute transport model was developed dispersivity is allowed to vary temporally as a function of the mean travel distance of the solute from an input source various scale dependent dispersivity relationships were presented including types which approach continuously to a maximum or asymptotic value the finite element model results were verified for a one dimensional case by comparison of the theoretical variances of the solute distribution for a given dispersivity travel distance function with the variances calculated from the finite element model concentration distributions the model was applied successfully to the results of several tracer tests reported in the literature that have exhibited a scale effect for systems that exhibit a constant asymptotic dispersivity at large times or mean travel distances the importance of scale dependent dispersion at early times or short travel distances was shown to be minimal in long term predictions of solute transport

void percolation and conduction of overlapping ellipsoids the void percolation and conduction problems for equisized overlapping ellipsoids of revolution are investigated using the discretization method the method is validated by comparing the estimated percolation threshold of spheres with the precise result found in literature the technique is then extended to determine the threshold of void percolation as a function of the geometric aspect ratio of ellipsoidal particles the finite element method is also applied to evaluate the equivalent conductivity of the void phase in the system the results confirm that there are no universalities for void percolation threshold and conductivity in particulate systems and these properties are clearly dependent on the geometrical shape of particles as a consequence void percolation and conduction associated with ellipsoidal particles of large aspect ratio should be treated differently from spheres

precise percolation thresholds of two dimensional random systems comprising overlapping ellipses abstract this work explores the percolation thresholds of continuum systems consisting of randomly oriented overlapping ellipses high precision percolation thresholds for various homogeneous ellipse systems with different aspect ratios are obtained from extensive monte carlo simulations based on the incorporation of vieillard baron s contact function of two identical ellipses with our efficient algorithm for continuum percolation in addition we generalize vieillard baron s contact function from identical ellipses to unequal ellipses and extend the monte carlo algorithm to heterogeneous ellipse systems where the ellipses have different dimensions and or aspect ratios based on the concept of modified excluded area a general law is verified for precise prediction of percolation threshold for many heterogeneous ellipse systems in particular the study of heterogeneous ellipse systems gains insight into the apparent percolation threshold symmetry observed earlier in systems comprising unequal circles consiglio et al 2004

inadmissibility of the maximum likelihood estimator in the sequential estimation of the size of a population summary for the sequential estimation of the size of a finite population the maximum likelihood estimator is a commonly used estimator but has no explicit expression in this paper it is shown that the maximum likelihood estimator is inadmissible an estimator which is simple in form and dominates the maximum likelihood estimator asymptotically is proposed to study the optimality property of the estimator for the stopping rule sb from a decision theoretical approach we investigate the admissibility of the estimator under a general weighted squared error loss function although the maximum likelihood estimator and its risk have no closed form expression the estimator will be shown to be inadmissible by means of a general theorem that we establish later as theorem 2 1 in this paper we only consider the simple capture capture model discussed by samuel 1968 more complex and realistic capture recapture models have been developed subsequently pollack otto 1983 pickands raghavachari 1987 bonett woodward bentler 1987 cormack 1989 huggins 1989 we conjecture that many of the

inference for the binomial n parameter a hierarchical bayes approach summary a hierarchical bayes approach to the problem of estimating n in the binomial distribution is presented this provides a simple and flexible way of specifying prior information and also allows a convenient representation of vague prior knowledge it yields solutions to the problems of interval estimation prediction and decision making as well as that of point estimation the bayes estimator compares favourably with the best previously proposed point estimators in the literature unknown parameters n and 0 most of the literature about statistical analysis of this model has focused on point estimation of n while interval estimation prediction and decision making have been little considered see 2 i adopt a hierarchical bayes approach this provides a simple way of specifying prior information and also allows a convenient representation of vague prior knowledge using limiting improper prior forms it leads to solutions of the problems of interval estimation prediction and decision making as well as that of point estimation a difficulty with bayesian analysis of this problem has been the absence of a sufficiently flexible and tractable family of prior distributions mainly due to the fact that n is an integer the present approach gets around this by first assuming that n has a poisson distribution the resulting hyperparameters are then continuous valued and one may use existing results about conjugate and vague priors in better understood settings

deterministic and stochastic analyses of dispersion in an unbounded stratified porous medium the dispersion of a conservative solute released instantaneously from a finite or point source in an unbounded nonrandom periodically stratified porous medium is examined theoretically by applying the moment method of r aris 1956 and p g saffman 1962 the governing moment equations are derived for a general stratified medium and then applied to study the detailed time dependent variations of various low order spatial moments of the three dimensional concentration distribution in a particular periodic sinusoidal stratified medium both for the general case when the mean flow direction is inclined to the stratification as well as for the special case of flow parallel to the stratification the present results confirm the previous results of v k gupta and k n bhattacharya 1986 regarding the asymptotic large time large scale dispersion coefficients for such a periodic medium the present results which are obtained through a deterministic analysis are compared also with the results which would be obtained by using the previous stochastic theories of g matheron and g demarsily 1980 and l w gelhar et al 1979 the comparison reveals several similarities as well as some important differences between the results of the deterministic and stochastic analyses

invasion percolation on substrates with correlated disorder abstract simulations have been carried out to investigate invasion percolation with trapping on two dimensional substrates with a multifractal distribution of bond threshold probabilities suprisingly the spatial correlations in the bond thresholds appear not to influence or influence only slightly the fractal scaling properties fractal dimensionalities d of about 1 82 for the entire clusters about 1 45 1 5 for the hull and about 1 4 1 45 for the outer hull were obtained for all bond invasion percolation models with and without spatial correlations the exponent describing the power law distribution of trapped defender cluster sizes has a value of about 1 90 d d d where d is the euclidean dimensionality the value for d of 1 4 1 45 for the outer hull is regarded as being consistent with but not in good agreement with the theoretical value of 4 3 for ordinary percolation the value of about 1 50 obtained for the total hull is much different from the value of 1 75 associated with ordinary percolation hulls the difference between the value of 1 82 obtained from the entire cluster and the theoretical percolation value of vl is 1 895833 is also regarded as being significant the simulation results also show that the fractal dimensionality for the total hull is much different for bond and site invasion percolation with trapping on substrates with uncorrelated disorder while the outer hulls have the same fractal dimensionality for both models for the site invasion percolation model d hull 7 4 while for the bond invasion percolation model d hull d outer hull

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

dispersion in fractal fracture networks the solute transport equation describes dispersion as a diffusion like process characterized by a constant dispersivity implying that small pulses of contaminant grow as the square root of distance traveled measured dispersivities tend to be proportional to the scale of the measurement suggesting that the pulses grow in linear proportion to distance an exponent r defined by the relationship d lr where d is the standard deviation of position of a contaminant particle and l is the distance traversed may be used to characterize dispersion it takes a value in the first case and 1 in the second the movement of a contaminant particle is analyzed here as a random walk through a fracture network composed of fissures of varying transmissivity if large transmissivity fractures are sufficiently few then one calculates r if large transmissivity fractures are numerous one obtains 2r q 1 where q is an exponent that describes the relationship between fracture spacing and transmissivity inserting the reasonable value q 1 gives r 1 this theory thus provides a physical basis for the observed scale dependence of dispersivity

nonuniversality and continuity of the critical covered volume fraction in continuum percolation we establish using mathematically rigorous methods that the critical covered volume fraction cvf for a continuum percolation model with overlapping balls of random sizes is not a universal constant independent of the distribution of the size of the balls in addition we show that the critical cvf is a continuous function of the distribution of the radius random variable in the sense that if a sequence of random variables converges weakly to some random variable then the critical cvf based on these random variables converges to the critical cvf of the limiting random variable

symmetry of the continuum percolation threshold in systems of two different size objects abstract we study the continuum percolation in systems composed of overlapping objects of two different sizes we show that when treated as a function of the volumetric fraction f as opposed to the concentration x the percolation threshold exhibits the symmetry c f r c 1 f r where r is the ratio of the volumes of the objects knowledge of this symmetry has the following benefits i the position of the maximum of the percolation threshold is then known to be at exactly f 1 2 for any r and ii full knowledge of the percolation threshold is obtained by performing simulations only for f 0 1 2 or f 1 2 1 whichever is computationally easier

on the erratic behavior of estimators of n in the binomial n p distribution abstract the problem of estimating the number of trials in a binomial distribution is known to be particularly difficult in certain circumstances both the maximum likelihood and method of moments estimators can produce very erratic results being extremely sensitive to minor fluctuations in the data but existing asymptotic theory for these estimators does not provide any insight into this peculiar behavior as the theory typically produces asymptotic normal distributions with respectable means and variances likewise the limiting distributions of other estimators do not convey much information about why those estimators might perform better than their more classical counterparts in this article we develop an alternative asymptotic theory that clearly and vividly depicts not only the erratic performance of the maximum likelihood and method of moments estimators but also the more regular robust properties of alternative estimators proposed by carroll and lombard we note that the parameter configurati

bayesian methods for prevalence estimation from incomplete administrative lists many studies aimed at estimating prevalence use several administrative lists from different sources in an attempt to enumerate all persons affected with the health condition of interest each list is incomplete in the sense that none of them enumerates all persons affected with the health condition further because the lists are drawn from different administrative sources the probability of enumeration varies from list to list the goal is to use information from the lists to estimate the total number of affected persons in the population but with some accounting for the different but unknown probabilities of enumeration on each list this paper presents a bayesian method to estimate prevalence when the probability of enumeration varies from list to list data from a survey of children with spina bifida illustrate the methodology

critical condition of the water retention model we study how much water can be retained without leaking through boundarieswhen each unit square of a two dimensional lattice is randomly assigned a blockof unit bottom area but with different heigh

fractal structure of equipotential curves on a continuum percolation model abstract we numerically investigate the electric potential distribution over a two dimensional continuum percolation model between the electrodes the model consists of overlapped conductive particles on the background with an infinitesimal conductivity using the finite difference method we solve the generalized laplace equation and show that in the potential distribution there appear quasi equipotential clusters which approximately and locally have the same values as steps and stairs since the quasi equipotential clusters have the fractal structure we compute the fractal dimension of equipotential curves and its dependence on the volume fraction over 0 1 the fractal dimension in 1 00 1 246 has a peak at the percolation threshold p c

universality and cluster structures in continuum models of percolation with two different radius distributions the percolation thresholds and the fractal cluster structures for continuum models of percolation with uniform cm1 and variable radius cm2 distributions of discs and spheres are investigated and compared with the results of ordinary lattice percolation configurations of up to 250000 discs 2 dimensions and 100000 spheres 3 dimensions are numerically simulated in two dimensions the authors find distinctly different percolation concentrationss for models cm1 and cm2 in the three dimensional systems the percolation concentration for both models cannot be distinguished within their limits of accuracy the fractal dimensions of the cluster hull surface and volume are the same as in the corresponding lattice models the harris criterion for the continuum percolation problem is confirmed by their simulation

analytical approximation of the two dimensional percolation threshold for fields of overlapping ellipses percolation of particle arrays is of high interest in microstructural design of materials while there have been numerous contributions to theoretical modeling of percolation in particulate systems no analytical approximation for the generalized problem of variable aspect ratio ellipses has been reported in the present work we 1 derive and verify through simulation an analytical percolation approach capable of identifying the percolation point in two phase materials containing generalized ellipses of uniform shape and size and 2 explore the dependence of percolation on the particle aspect ratio we validate our technique with simulations tracking both cluster sizes and percolation status in networks of elliptical and circular particles we also outline the steps needed to extend our approach to three dimensional particles ellipsoids for biological materials we ultimately aim to provide direct insight into the contribution of each single phase in multiphase tissues to mechanical or conductive properties for engineered materials we aim to provide insight into the minimum amount of a particular phase needed to strongly influence properties

optimization of shape in continuum percolation we consider a version of the boolean or poisson blob continuum percolation model where at each point of a poisson point process in the euclidean plane with intensity a a copy of a given compact convex set a with fixed rotation is placed to each a we associate a critical value c a which is the infimum of intensities a for which the occupied component contains an unbounded connected component it is shown that min c a a convex of area a is attained if a is any triangle of area a and max c a a convex of area a is attained for some centrally symmetric convex set a of area a it turns out that the key result which is also of independent interest is a strong version of the difference body inequality for convex sets in the plane in the plane the difference body inequality states that for any compact convex set a 4 a a a 6 a with equality to the left iff a is centrally symmetric and with equality to the right iff a is a triangle here denotes area and a a is the difference body of a we strengthen this to the following result for any compact convex set a there exist a centrally symmetric convex set c and a triangle t such that c t a and c c a t t with equality to the left iff a is centrally symmetric and to the right iff a is a triangle

estimation of percolation thresholds via percolation in inhomogeneous media this paper mathematically justifies techniques used to estimate the percolation thresholds of fully penetrable disks or boolean models of planar disks generalizations to systems of other particles in two or more dimensions are also discussed

asymmetry in the percolation thresholds of fully penetrable disks with two different radii we perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii as a function of volumetric proportion v the percolation threshold phic v lambda is approximately symmetric about v 0 5 for a fixed ratio lambda of the radii however the difference from symmetry is strongly statistically significant we also improve by an order of magnitude the measurement of the percolation threshold for disks of equal radius phic 0 676 347 5 6

stabilizing binomial n estimators abstract the maximum likelihood estimator of the binomial parameter n can be highly unstable resulting in huge fluctuations in the estimate when the data are only slightly perturbed a method of assessing this sensitivity based on perturbing the log likelihood function is proposed and examined

a comparison of n estimators for the binomial distribution abstract estimating the number n of trials given a sequence of independent success counts obtained by replicating the n trial experiment is less studied and a considerably harder problem than estimating p for the binomial distribution both the method of moments and the maximum likelihood estimators of n are considered when p is unknown and small while n is large both estimators become highly unstable stabilized versions of these estimators are proposed and their performance is described in terms of results obtained by simulation

stochastic modeling of groundwater flow by unconditional and conditional probabilities 2 the solute transport the problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two and three dimensional flows approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity the solute concentration regarded as a random variable is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty the uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale in any case the concentration expectation does not obey a diffusion type equation in the case of two dimensional flows unless the solute body has traveled a distance larger than a few tens transmissivity integral scales this distance may be exceedingly large in many conceivable applications

stochastic theory of field scale fickian dispersion in anisotropic porous media a three dimensional theory is described for field scale fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities the study relies partly on earlier work by the authors the attributes of which are briefly reviewed it leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media we express the dispersion tensor d as the sum of a local component d and a field scale component the local component is assumed to be independent of velocity which is most appropriate if it represents molecular diffusion and its principal terms are taken to act parallel and normal to the mean velocity vector the field scale component is written as where is a dispersivity tensor and we show that at large peclet numbers p the dispersivity tensor reduces to a single principal component parallel to the mean velocity regardless of how is oriented this result valid for arbitrary covariance functions of log hydraulic conductivity differs from that of l w gelhar and c l axness 1983 according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value they calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation or away from the stratification in typical layered media we show that this principal dispersivity is offset in the opposite direction at small and intermediate peclet numbers but rotates toward the mean velocity as p increases the largest eigen value is constant and dominated by field scale velocity fluctuations at large p values the other two eigen values diminish asymptotically in proportion to p 1 and are controlled by d as well as by field scale differential convection the range of small peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low permeability rocks or soils we show that at low p values all three principal dispersivities are proportional to p and thus is proportional to 2 a phenomenon reminiscent of taylor diffusion when the mean velocity is inclined to the axes of anisotropy the eigen values of are neither parallel nor normal to however since d is dominated by d at small peclet numbers the principal dispersion coefficients are asymptotically as p 0 parallel and normal to the mean velocity just like when p is large their maximum deviation from these directions occurs at intermediate p values

efficient measurement of the percolation threshold for fully penetrable discs we study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs we use two different algorithms to efficiently simulate the frontier including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice we find that c 0 676 339 0 000 004 thus providing an extra significant digit of accuracy to this constant some figures in this article appear in colour in the electronic version see www iop org

stochastic subsurface hydrology from theory to applications research on stochastic analysis of subsurface flow has developed rapidly in the last decade but applications of this approach have been very limited the purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large scale behavior of naturally heterogeneous aquifers perturbation based spectral theory which presumes local statistical homogeneity provides generic theoretical results for the head variance effective conductivity tensor and macrodispersivity tensor in a field these results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field field information of variances and correlation scales of natural materials is summarized the validity of some of the generic stochastic results is evaluated through comparisons with monte carlo simulations and field observations a specific field application example is developed to illustrate how the stochastic results are used to estimate large scale parameters and determine the reliability of three dimensional numerical simulations using typical log conductivity covariance parameters the effective hydraulic conductivity tensor and the macrodispersivity tensor are estimated the calculated head variance based on the simulated mean hydraulic gradient is used as a measure of the adequacy of the calculation of the steady state flow model discussion emphasizes limitations and extensions of this approach and ongoing field evaluations of the results

on bayesian estimation of group size summary in estimating the size of a group of animals it is commonly the case that the sighting probability for individuals is less than one if each individual in the group is independently sighted with equal probability the problem is one of estimating the number of trials from the number of successes in one or more binomial experiments in many cases some individuals in the group have unit sighting probability whereas others do not this note describes the bayesian estimation of group size for this case