subordinated advection dispersion equation for contaminant transport a mathematical method called subordination broadens the applicability of the classical advection dispersion equation for contaminant transport in this method the time variable is randomized to represent the operational time experienced by different particles in a highly heterogeneous aquifer the operational time captures the fractal properties of the medium this leads to a simple parsimonious model of contaminant transport that exhibits many of the features heavy tails skewness and non fickian growth rate typically seen in real aquifers we employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives applied to a one dimensional approximation of the made 2 data set the model shows excellent agreement

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

governing equations and solutions of anomalous random walk limits fachbereich mathematik university of dortmund 44221 dortmund germany received 5 june 2002 published 30 december 2002 continuous time random walks model anomalous diffusion coupling allows the magnitude of particlejumps to depend on the waiting time between jumps governing equations for the long time scaling limits ofthese models are found to have fractional powers of coupled space and time differential operators explicitsolutions and scaling properties are presented for these equations which can be used to model ow in porousmedia and other physical systems doi 10 1103 physreve 66 060102 pacs number s 05 40 2a 02 50 cw 05 60 2k 47 55 mhi introduction

asymptotic distributions of continuous time random walks a probabilistic approach we provide a systematic analysis of the possible asymptotic distributions o one dimensional continuous time random walks ctrws by applying the limit theorems of probability theory biased and unbiased walks of coupled and decoupled memory are considered in contrast to previous work concerning decoupled memory and levy walks we deal also with arbitrary coupled memory and with jump densities asymmetric about its mean obtaining asymmetric levy stable limits suprisingly it is found that in most cases coupled memory has no essential influence on the form of the limiting distribution we discuss interesting properties of walks with an infinite mean waiting time between successive jumps

transport behavior of a passive solute in continuous time random walks and multirate mass transfer 1 we consider the transport behavior of a passive solute in a heterogeneous medium modeled by continuous time random walks ctrw and linear multirate mass transfer mrmt within the ctrw framework we formulate a transport model which is formally equivalent to mrmt in both approaches the total concentration is divided into mobile and immobile parts the immobile concentration is given by the convolution in time of the mobile concentration and a memory function the memory function is a functional of the distribution of transition and trapping times for the ctrw and the mrmt approach respectively and determines the transport behavior of the solute based on different expressions for the memory function in the two frameworks we derive conditions for which both approaches describe the same transport behavior we focus on anomalous transport behavior that can arise if the transition and trapping time distributions behave algebraically in a given time regime using an expansion of the laplace transform of the memory function we develop explicit expressions for the time behavior of the flux concentration as well as for the center of mass velocity and the macro dispersion coefficients of the solute distribution we observe anomalous power law as well as normal fickian transport behavior depending on the exponents that dominate the trapping and transition time distributions respectively the results show that the character of the anomalous transport does not depend on the details of the transport model but only on the exponents dominating the transition or trapping time distributions the unified transport framework presented here comprising ctrw and mrmt shows new aspects and opens new perspectives for the modeling of transport in heterogeneous media

stochastic solution of space time fractional diffusion equations classical and anomalous diffusion equations employ integer derivatives fractional derivatives and other pseudodifferential operators in space in this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are mittag leffler type this leads to explicit solutions for space time fractional diffusion equations with multiscaling space fractional derivatives and additional insight into the meaning of these equations

on the role of diffusivity in some stable equilibria of a diffusion equation our main goal in this work is to exhibit a class of diffusivity functionswhich regardless of the shape of 0 allows for the existence of patternsfor p by considering p as a singularly perturbed problem where the dif fusivity is given by k x a small positive parameter the existence of pat terns which develop boundary transition layers and spikes was studied in n1 the case in which these patterns develop internal transition layerswas studied for a more general equation in n2 using an approach basedon 1 convergence see also n3 n4 problem p was considered in m for the case in which k is a constantfunction therein the author resorted to the shape of 0 to produce pat terns therefore the geometric structure of the patterns found here is dif ferent from the geometric structure of the ones found in m actually the technique we use here stems from the one presented in m where after finding a invariant set for the flow the monotonicity propertyof the flow as well as property s see m p 431 which is based onzorn s lemma were used our approach is different since after finding an invariant set we only usebasic results for the flow and variational techniques to show existence of alocal minimiser of the energy functional in this invariant set this localminimiser turns out to be a stable stationary solution to p the very same analysis still go through for the case in which the trivialequilibrium solutions a and b are positive i e 0 a b and the reactionterm is replaced by a spatially inhomogeneous term s x f v where s x isa positive continuous function problem p appears as a mathematical model in many distinct areasand just for the sake of illustration as well as interpreting the results froma physical point of view let us suppose that v models the time evolutionof the concentration of a diffusing substance in a medium whose diffusivityis given by k x x 0 and under the effect of a source term f when0 a b then typically as far as the existence of patterns in higher space dimen sion is concerned the diffusivity is small on a closed surface s 0 but thisis not sufficient and another condition on the behavior of k x has to beimposed in n2 n4 this condition relates the slope and concavity of k x alongthe normal to s to the gaussian and mean curvatures of s herein as longas the areas of the negative and positive bumps of f are the same theequal area condition it is only required that k x be large on two smootharbitrary sets o

on the existence of multiple ordered solutions of nonlinear eigenvalue problems on etudie le probleme de dirichlet u f u dans u o sur ou est un domaine borne lisse de r n n 1 la fonction a valeur reelle f o r est supposee c 1 et o est un parametre reel on etudie le lien entre les changements de signe de la fonction f et l existence de plusieurs solutions positives ordonnees

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

progressive image transmission using vector quantization on images in pyramid form a progressive image transmission scheme in which vector quantization is applied to images represented by pyramids is proposed a mean pyramid representation of an image is first built up by forming a sequence of reduced size images by averaging over blocks of 2 2 pixels a difference pyramid is then built up by taking the differences between successive levels in the mean pyramid progressive transmission is achieved by sending all the nodes in the difference pyramid starting from the top level and ending at the bottom level the kth approximate image can be formed by adding the information of level k to the previously reproduced k 1 st approximation to gain efficiency vector quantization is applied to the difference pyramid of the image on a level by level basis if the errors due to quantization at level k are properly delivered and included in the next level k 1 then it is demonstrated that the original image can be reconstructed an entropy coder is used to encode the final residual error image losslessly thus ensuring perfect reproduction of the original image the experiments demonstrate that it is possible to achieve simultaneously lossless and progressive transmission with compression at the intermediate level the use of vector quantization results in a coding gain over that obtained using only a huffman coder excellent reproduction is achieved at a bit rate of only 0 06 bits pixel

bifurcations in a parabolic equation with variable diffusion on considere le probleme aux valeurs limites u t a 2 x u x x f u 0 x 1 u x 0 t u x 1 t 0 pour 0 f u u u 3 a x 1 x et a x x 0 1 on etudie la stabilite et la bifurcation des solutions d equilibre pour les cas x 0 1 0 1 et x 0 1 2 0 1 2 et a x a 1 x pour x 1 2 1

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

theory of anomalous chemical transport in random fracture networks we show that dominant aspects of chemical particle transport in fracture networks char22 non gaussian propagation char22 result from subtle features of the steady flow field distribution through the network this is an outcome of a theory based on a continuous time random walk formalism structured to retain the key space time correlations of particles as they are advected across each fracture segment the approach is designed to treat the complex geometries of a large variety of fracture networks and multiscale interactions monte carlo simulations of steady flow in these networks are used to determine the distribution of velocities in individual fractures as a function of their orientation the geometry and velocity distributions are used in conjunction with particle mixing rules to map the particle movement between fracture intersections onto a joint probability density ensuremath psi mathbf r t the chemical concentration plume and breakthrough curves can then be calculated analytically particle tracking simulations on these networks exhibit the same non gaussian profiles demonstrating quantitative agreement with the theory the analytic plume shapes display the same basic behavior as extensive field observations at the columbus air force base mississippi the quantitative correlation between the time dependence of the mean and standard deviation of the field plumes and their shape is predicted by the theory

comparative performance of pyramid data structures for progressive image transmission a review of various pyramid data structures for progressive image transmission focusing on the steps involved in forming the pyramid and the performance comparisons is presented the simple mean pyramid its variants including the truncated mean reduced sum difference reduced difference and s transform pyramids and the more general laplacian pyramid are discussed it is also shown that the use of an interpolation function improves the quality of the full resolution approximations both objectively and subjectively

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in n dimensional domains we prove existence and establish the asymptotic behavior as e 0 of stable stationary solutions to the equation ut e d x u 1 u2 u a x for t x r where rn n 2 with neumann boundary condition the function a x c0 satisfies 1 a x 1 and vanishes on some hypersurfaces the results generalize to n dimensional domains and to variable diffusivity earlier paper by angenent et al j differential equations 67 1987 212

stability results for a reaction diffusion problem with mixed boundary conditions and applications to some symmetric cases abstract in this article we consider a one dimensional reaction diffusion problem with mixed boundary conditions we provide conditions for the existence or nonexistence of stable nonconstant solutions whose derivative vanishes at some point as an application we obtain similar results for problems with dirichlet boundary conditions posed in some symmetric domains an n dimensional ball surfaces of revolution and model manifolds

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

fractal mobile immobile solute transport 1 a fractal mobile immobile model for solute transport assumes power law waiting times in the immobile zone leading to a fractional time derivative in the model equations the equations are equivalent to previous models of mobile immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times the solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase in this regard the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase the solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers including power law breakthrough curves at late time and power law decline in the measured mobile mass the made site mobile tritium mass decline is consistent with a fractional time derivative of order 0 33 while haggerty et al s 2002 stream tracer test is well modeled by a fractional time derivative of order 0 3

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

deterministic prediction in progressive coding deterministic prediction in progressive coding of images is investigated progressive coding first creates a sequence of resolution layers by beginning with an original image and reducing its resolution several times by factors of some natural number m the resultant layers are losslessly encoded beginning with the lowest resolution layer and then encoding each higher resolution image incrementally upon the previous one coding efficiency may be improved if knowledge of the rules which produced the lower resolution image of each pair is used to deterministically predict pixels of the higher so they need not be encoded given reduction rules expressing each low resolution pixel as a function of nearby high resolution pixels and previously generated low resolution pixels it is shown that finding a complete set of rules each of which deterministically predicts the value of a high resolution pixel when certain values are found in nearby low resolution pixels and previously coded high resolution pixels is np complete a recursive algorithm for solving the problem in optimal time as a depth first tree search is proposed and the characteristics of the resultant prediction process are studied

structure of the set of steady state solutions and asymptotic behaviour of semilinear heat equations soit un domaine borne lisse connexe de r n on considere a la fois le probleme aux valeurs initiales u t u f u dans 0 u x t 0 sur 0 u x o u 0 x dans et le probleme stationnaire u g u dans u 0 sur ou u 0 est une condition initiale donnee et f une non linearite donnee de classe c 1

on advective transport in fractal permeability and velocity fields we consider advective transport in a steady state random velocity field with homogeneous increments such a field is self affine with a power law dyadic semivariogram s proportional to d2 where d is distance and is a hurst coefficient it is characterized by a fractal dimension d e 1 where e is topological dimension as the mean and variance of such a field are undefined we condition them on measurement at some point x0 we then introduce a tracer at another point y0 and invoke elementary theoretical considerations to demonstrate that its conditional mean dispersion is local at all times its conditional mean concentration and variance are given explicitly by well established expressions which however have not been previously recognized as being valid in fractal fields once the conditional mean travel distance s of the tracer becomes large compared to the distance between y0 and x0 the corresponding dispersion and dispersivity tensors grow in proportion to s1 2 where 0 1 this supralinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained by standard methods of interpretation from tracer behavior observed in a variety of geologic media under varied flow and transport regimes filtering out modes from the fractal velocity field with correlation scales larger than some s0 allows an asymptotic transport regime to develop when s s0 the corresponding asymptotic dispersivities grow in proportion to s02 when 0 this linear to sublinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained from calibrated numerical models in a variety of media a self affine natural log permeability field gives rise to a self affine velocity field while s is sufficiently small to insure that the variance of the log permeabilities which grows as a power of s remains nominally less than one an analysis of published apparent longitudinal dispersivity data in light of the above theoretical results supports my earlier conclusion that when one juxtaposes data from a large number of generally dissimilar geologic media from a variety of locales one observes a tendency toward self affine behavior with a hurst coefficient 0 25 at any given locale such media may or may not exhibit fractal behavior if they do may or may not be close to 0 25

the roles of diffusivity and curvature in patterns on surfaces of revolution abstract we address the question of finding sufficient conditions for existence as well as nonexistence of nonconstant stable stationary solution to the diffusion equation u t div a u f u on a surface of revolution with and without boundary conditions found relate the diffusivity function a and the geometry of the surface where diffusion takes place in the case where f is a bistable function necessary conditions for the development of inner transition layers are given

possible problems of scale dependency in applications of the three dimensional fractional advection dispersion equation to natural porous media 1 a three dimensional 3 d analysis of transport and macrodispersion at the macrodispersion experiment made site boggs et al 1993 using the fractional advection dispersion equation fade developed by meerschaert et al 1999 2001 shows that the levy dispersion process is scale dependent levy dispersion may be superior to gaussian dispersion on a sufficiently small scale on larger scales both theories are likely to suffer from the fact that because of depositional structures most flow fields display an evolving nonstationary structure motion in such fields is advection dominated displays a lot of memory and therefore is not modeled well by markov random processes which underlie the derivation of both the gaussian and levy advection dispersion equations berkowitz et al 2002 to improve plume simulation of an advection dominated transport process one would have to bring in more advective irregularity while simultaneously decreasing the levy dispersion coefficient therefore on a 3 d basis first order levy dispersion has limitations similar to gaussian dispersion however this and related theories such as the continuous time random walk ctrw formalism are in the early stages of development and thus may be fruitful areas for further research