quantum function algebra at roots of 1 abstract we introduce a form of the quantum function algebra on a drinfeld jimbo quantum group over the ring z q q 1 specializing q to a root of 1 we show that over the cyclotomic field this algebra is a projective module over its central sub algebra which is the usual coordinate algebra of the group we study the induced poisson lie structure of the group a bundle of algebras on a complex simply connected lie group with hamiltonian flows in the bundle is constructed some representations of the quantum function algebra in a root of 1 are constructed as an application an estimate of the dimension of an arbitrary representation is given

two splitting positive definite mixed finite element methods for parabolic integro differential equations abstract two novel mixed finite element procedures are established for parabolic integro differential equations which can be split into two independent symmetric positive definite sub schemes and do not need to solve a coupled system of equations without requiring the lbb consistency condition the convergence analysis shows that both methods lead to the optimal order l 2 norm error estimate for u and optimal h div norm error estimate for a numerical example is presented to illustrate the theoretical analysis

double partition quantum cluster algebras abstract a family of quantum cluster algebras is introduced and studied in general these algebras are new but sub classes have been studied previously by other authors the algebras are indexed by double partitions or double flag varieties equivalently they are indexed by broken lines l by grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of one broken line to another compatible pairs can be written down the algebras are equal to their upper cluster algebras the variables of the quantum seeds are given by elements of the dual canonical basis

modeling flow and transport in highly heterogeneous three dimensional aquifers ergodicity gaussianity and anomalous behavior 2 approximate semianalytical solution 1 flow and transport take place in a heterogeneous medium of lognormal distribution of the conductivity k flow is uniform in the mean and the system is defined by u mean velocity y2 log conductivity variance and integral scale i transport is analyzed in terms of the breakthrough curve of the solute identical to the traveltime distribution at control planes at distance x from the source the self consistent approximation is used where the traveltime is approximated by the sum of pertinent to the different separate inclusions and the neglected interaction between inclusions is accounted for by using the effective conductivity the pdf f x where x is the control plane distance is derived by a simple convolution it is found that f x has an early arrival time portion that captures most of the mass and a long tail which is related to the slow solute particles that are trapped in blocks of low k the macrodispersivity is very large and is independent of x the tail f x is highly skewed and only for extremely large x i depending on y2 the plume becomes gaussian comparison with numerical simulations shows very good agreement in spite of the absence of parameter fitting it is found that finite plumes are not ergodic and a cutoff of f x is needed in order to fit the mass flux of a finite plume depending on y2 and x i the bulk of f x can be approximated by a gaussian shape with fitted equivalent parameters the issue of anomalous behavior is examined with the aid of the model

statistical theory of groundwater flow and transport pore to laboratory laboratory to formation and formation to regional scale the statistical approach has been applied increasingly to groundwater flow problems in the last decade as is illustrated in figure 3 by the cumulative number of articles published in this field in water resources reseach this development has been motivated by the recognition of the fact that porous formations are heterogeneous i e with properties which vary in an irregular manner in space flow domains are characterized by the length scale l of their spatial extent and three such scales of a fundamental nature are introduced the laboratory the local and the regional scale heterogeneity is characterized by the spatial correlation scale i of the property of interest the three scales corresponding to the above ones being the pore scale the log hydraulic conductivity and the log transmissivity integral scales the medium properties and related flow variables are regarded as random space functions which satisfy two basic requirements they enjoy some type of stationarity and i l an additional scale d is the measurement or computational scale characterizing the size of the measurement device of a flow variable or the element over which the variable is averaged for computational purposes in both cases the interest resides in the space average of the flow variable over a volume or area of length scale d the primary aim of the theory of flow and transport through porous media is to determine the statistical moments of the space averaged variables given the statistical structure of the spatially variable property the main objective of the study is to show that flow and transport problems at the three fundamental scales can be treated by a unified statistical approach along this line the specific aspects of each scale are examined separately and areas of interest for future research are indicated in the concluding remarks it is submitted that the statistical approach to groundwater flow has become a comprehensive theory beyond the stage of an ad hoc modeling technique

stochastic modeling of macrodispersion in heterogeneous porous media a method for the prediction of dispersion processes of inert solutes occurring in heterogeneous porous media is presented its main features are as follows 1 the velocity field is assumed to be a space random function its moments are expressed through the physical parameters of the log transmissivity and head random fields by linearizing the flow equation the random velocity being a linear function of the head and log transmissivity which are assumed to be jointly multivariate normal mvn is thus also mvn its pdf is completely defined by its first two moments 2 the statistics of the dispersion process are obtained by the particle tracking method through monte carlo simulations which are based on gaussian conditioning the method is applied to the borden natural gradient tracer test freyberg 1986 showing favorable results

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

solute transport in highly heterogeneous aquifers the paper deals with the transport of nonreactive solute in heterogeneous formations with prescribed statistical properties of the hydraulic log conductivity y ln k available solutions obtained from perturbation methods are limited to first and second order solutions valid only for small values of the log conductivity variance 2y published numerical investigations give comparative results with finite values of 2y but some discrepancies among the results generate doubts about the capability of numerical methods to capture high order effects for media with large variance values when these large 2y values are encountered in natural formations the related nonlinear effects could be significant in the velocity statistics and in the overall dispersion process the nonlinearity consequences are here investigated in two dimensional isotropic porous media by the monte carlo technique coupled with a finite element analysis the analysis includes 2y values from 0 05 to 4 enhancing the relevance of the nonlinear effects in the dispersion tensor solution to dissipate the doubts related to the numerical approach the accuracy of the solutions was defined by checking the influence of the factors which can affect the solution and by giving an estimation of the related errors the numerical results confirm the validity of the first order and second order analyses when 2y 0 the second order solution captures the nonlinear effects in a small range of log conductivity variance close to zero the nonlinear terms neglected in the first order formulation for higher 2y values give 1 late travel time longitudinal dispersion values greater than the linear solution 2 notably non gaussian distribution of the lagrangian velocity and particle displacements and 3 travel times to approach the asymptotic fickian regime longer than those obtained using the linear solution

a class of quadratic matrix algebras arising from the quantized enveloping algebra uq a2n 1 a natural family of quantized matrix algebras is introduced it includes the two best studied such located inside uq a2n 1 it consists of quadratic algebras with the same hilbert series as polynomials in n2 variables we discuss their general properties and investigate some members of the family in great detail with respect to associated varieties degrees centers and symplectic leaves finally the space of rank r matrices becomes a poisson submanifold and there is an associated tensor category of rank r matrices

eulerian lagrangian theory of transport in space time nonstationary velocity fields exact nonlocal formalism by conditional moments and weak approximation a unified eulerian lagrangian theory is presented for the transport of a conservative solute in a random velocity field that satisfies locally v x t f x t where f x t is a random function including sources and or the time derivative of head solute concentration satisfies locally the eulerian equation c x t t j x t g x t where j x t is advective solute flux and g x t is a random source independent of f x t we consider the prediction of c x t and j x t by means of their unbiased ensemble moments c x t and j x t conditioned as implied by the subscript on local hydraulic measurements through the use of the latter in obtaining a relatively smooth unbiased estimate x t of v x t these predictors satisfy c x t v t j x t g x t where j x t x t c x t q x t and q x t is a dispersive flux we show that q is given exactly by three space time convolution integrals of conditional lagrangian kernels with q with c and with c for a broad class of v x t fields including fractals this implies that q x t is generally nonlocal and non fickian rendering c x t non gaussian the direct contribution of random variations in f to q depends on c rather than on c we elucidate the nature of the above kernels discuss conditions under which the convolution of and c becomes pseudo fickian with a lagrangian dispersion tensor similar to that derived in 1921 by taylor recall a 1952 result by batchelor which yields an exact expression for c at early time use the latter to conclude that linearizations which predict that c bifurcates at early time when the probability density function of v is unimodal cannot be correct propose instead a weak approximation which leads to a nonlinear integro differential equation for c due to an instantaneous point source and which improves with the quantity and quality of hydraulic data demonstrate that the weak approximation is analogous to the direct interaction closure of turbulence theory offer non fickian and pseudo fickian weak approximations for the second conditional moment of the concentration prediction error demonstrate that it yields the so called two particle covariance as a special case conclude that the conditional variance of c does not become infinite merely as a consequence of disregarding local dispersion and discuss how to estimate explicitly the cumulative release of a contaminant across a compliance surface together with the associated estimation error

quantized rank r matrices abstract first some old as well as new results about p i algebras ore extensions and degrees are presented then quantized n r matrices as well as certain quantized factor algebras m r 1 q n of m q n are analyzed for r 1 n 1 m r 1 q n is the quantized function algebra of rank r matrices obtained by working modulo the ideal generated by all r 1 r 1 quantum subdeterminants and a certain localization of this algebra is proved to be isomorphic to a more manageable one in almost all cases the quantum parameter is a primitive m th root of unity the degrees and centers of the algebras are determined when m is a prime and the general structure is obtained for arbitrary m

on diffusion in fractal porous media generalized hydrodynamics and irreversible thermodynamics are used to derive the relation between a general wave vector and frequency dependent diffusion tensor and the power spectral density the results are valid for porous media with evolving heterogeneities an integrodifferential transport equation valid in porous media with evolving heterogeneities is presented

a note on head and velocity covariances in three dimensional flow through heterogeneous anisotropic porous media a procedure for the characterization of the head and velocity fields for saturated flow in heterogeneous statistically anisotropic formations is developed the fields are characterized through a series of spatial covariances as well as the velocity head velocity log conductivity and head log conductivity cross covariances all these functions are given in terms of the parameters which characterize the input log conductivity field and the average head garadient a general procedure is outlined followed by explicit derivation of head and velocity covariances for heterogeneous statistically anisotropic formations closed form analytic results are obtained for the statistically isotropic formations while the results for the anisotropic ones required taking quadrature over analytic expressions the results may be used for transport predictions and for conditioning on measurements and may be particularly useful for testing of numerical codes

expansions in statistical physics on presente un traitement auto contenu du developpement polymere on axiomatise cette notion on demontre comment ces developpements permettent un traitement systematique de la plupart des developpements en amas de la physique statistique

transport in multiscale log conductivity fields with truncated power variograms di federico and neuman this issue investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic gaussian fields each associated with an exponential autocovariance here we present leading order expressions for the displacement covariance and dispersion coefficient of an ensemble of solute particles advected through such a flow field in two or three dimensions both quantities are functions of the mean travel distance s the hurst coefficient h and the low and high frequency cutoff integral scales l and u the latter two are related to the length scales of the sampling window region under investigation and sample volume data support respectively if one considers transport to be affected by a finite domain much larger than the mean travel distance so that s l then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s if one considers transport to be affected by a domain which increases in proportion to s then l and s are of similar order and a preasymptotic regime never develops instead transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale l s we show that if additionally u l then the corresponding longitudinal dispersivity grows in proportion to l1 2h or equivalently s1 2h both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal fickian dispersivities with scale we conclude by investigating the effect of variable separations between cutoff scales on dispersion

on transport in porous formations characterized by heterogeneity of evolving scales solute transport in natural formations at the regional scale is influenced by several scales of heterogeneity which correspond to the presence of several geological units called facies as customarily assumed in stochastic theories inside the facies the transport can be characterized by a single scale of heterogeneity at the regional scale several geological units are present such that a hierarchy of relevant scales needs to be defined a possible model for this spatial variability assumes the log conductivity as a random space function of stationary increments characterized by a power law semivariogram with this hypothesis the ergodic dispersion coefficient grows unbounded as time increases leading to the phenomenon called anomalous dispersion an alternative approach considers the plume in nonergodic conditions and assumes the effective dispersion coefficient which is defined through differentiation in time of the expected value of the spatial second order plume moment as representative of macrodispersion large differences have been observed in the resulting plume spreading while approaching the problem using the above alternative definitions in this paper we provide first order analytical solutions for the longitudinal effective dispersion coefficient d l as well as for the expected value of the longitudinal spatial plume moment that complement semianalytical expressions recently proposed in literature furthermore we provide a semianalytical expression for the standard deviation of the longitudinal second order moment which is important in assessing the interval of confidence of the estimation provided by suitable numerical simulations are performed to validate analytical and semianalytical expressions as well as to assess the impact of the cutoff in the log conductivity power spectrum imposed by choosing a finite domain dimension we conclude that according to recently published results the dispersion is anomalous when the hurst coefficients h is larger than 0 5 while it is fickian for h 0 5 this is in contrast with the ergodic analysis which concludes that the dispersion is anomalous irrespective of the hurst coefficient hence the effective dispersion coefficient is more effective than the ergodic dispersion coefficient to represent the plume spreading however the standard deviation of the longitudinal spatial second order moment is of the same order of magnitude as the expected value leading to the conclusion that the estimations provided by d l and are affected by large uncertainties numerical results are in good agreement with the analytical solutions and under some hypotheses they are not influenced by the cutoff this is not the case for the ergodic second order longitudinal moment which strongly depends on the imposed cutoff

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

incremental closure for systems of two variables per inequality subclasses of linear inequalities where each inequality has at most two vari ables are popular in abstract interpretation and model checking because they strike a balance between what can be described and what can be efficiently computed this paper focuses on the tvpi class of inequalities for which each coefficient of each two variable inequality is unrestricted an implied tvpi in equality can be generated from a pair of tvpi inequalities by eliminating a given common variable echoing resolution on clauses this operation called result can be applied to derive tvpi inequalities which are entailed implied by a given tvpi system the key operation on tvpi is calculating closure satisfiability can be observed from a closed system and a closed system also simplifies the calculation of other operations a closed system can be derived by repeatedly applying the result operator the process of adding a single tvpi inequality to an already closed input tvpi system and then finding the closure of this augmented system is called incremental closure this too can be calcu lated by the repeated application of the result operator this paper studies the calculus defined by result the structure of result derivations and how deriva tions can be combined and controlled a series of lemmata on derivations are presented that collectively provide a pathway for synthesising an algorithm for incremental closure the complexity of the incremental closure algorithm is analysed and found to be o n 2 m 2 lg m where n is the number of variables and m the number of inequalities of the input tvpi system

temporal scaling of hydraulic head fluctuations nonstationary spectral analyses and numerical simulations 1 temporal scaling of the hydraulic head time series h t was found in a previous analysis of hourly measured head data this issue is further investigated in this paper with nonstationary spectral analyses and numerical simulations the results show that temporal scaling may indeed exist in h t which fluctuates like a fractional brownian motion in most aquifers on the basis of a linear reservoir model with a white noise recharge input we show that the variance and covariance of h t are functions of time the head variance increases with time and approaches a constant limit as time progresses while the covariance decreases with the separation time interval for a fixed time and approaches the typical exponential covariance as time increases the spectra of the simulated h t using a one dimensional transient groundwater flow model with a white noise recharge in both homogeneous and heterogeneous aquifers are shown to be proportional to f where f is frequency and 1 84 or h 0 42 heterogeneity in the hydraulic conductivity may affect the fractal dimension of h t in highly permeable aquifers but not in the low permeable aquifer simulated in this study

universal scaling of hydraulic conductivities and dispersivities in geologic media an interpretation is offered for the observation that dispersivities increase with scale apparent longitudinal dispersivity data from a variety of hydrogeologic settings are assumed to represent a continuous hierarchy of log hydraulic conductivity fields with mutually uncorrelated increments each field having its own exponential autocovariance associated integral scale and variance that increases as a power of scale such a hierarchy is shown theoretically to form a self similar random field with homogeneous increments regardless of whether or not the underlying assumption is valid one can justify interpreting the apparent dispersivities in a manner consistent with a recent quasi linear theory of non fickian and fickian dispersion in homogeneous media which supports the notion of a self similar hierarchy a posteriori the hierarchy is revealed to possess a semivariogram s cs where c is a constant and a fractal dimension d e 0 75 where e is the topological dimension of interest this can be viewed as a universal scaling rule about which large deviations occur due to local influences including the existence of discrete natural scales at which log hydraulic conductivity is statistically homogeneous as such homogeneity is at best a local phenomenon occurring intermittently over narrow bands of the scale spectrum one must question the utility of associating medium properties with representative elementary volumes and relying on fickian models of dispersion over more than relatively narrow scale intervals porous and fractured media appear to follow the same idealized scaling rule for both flow and transport raising a question about the validity of many distinctions commonly drawn between such media finally the data suggest that conditioning transport models through calibration against hydraulic measurements has the effect of filtering out large scale modes from the hierarchy

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

a random field model for anomalous diffusion in heterogeneous porous media heterogeneity as it occurs in porous media is characterized in terms of a scaling exponent or fractal dimension a feature of primary interest for two phase flow is the mixing length this paper determines the relation between the scaling exponent for the heterogeneity and the scaling exponent which governs the mixing length the analysis assumes a linear transport equation and uses random fields first in the characterization of the heterogeneity and second in the solution of the flow problem in order to determine the mixing exponents the scaling behavior changes from long length scale dominated to short length scale dominated at a critical value of the scaling exponent of the rock heterogeneity the long length scale dominated diffusion is anomalous

the center of the quantized matrix algebra abstract we compute the center in the case where q is a root of unity the main steps are to compute the degree of an associated quasipolynomial algebra and to compute the dimensions of some interesting irreducible modules

non fickian convection diffusion models in porous media in this paper we propose a numerical scheme to approximate the solution of a non fickian coupled model that describes e g miscible transport in porous media the model is defined by a system of a quasilinear elliptic equation which governs the fluid pressure and a quasilinear integro differential equation which models the convection diffusion transport process the numerical scheme is based on a conforming piecewise linear finite element method for the discretization in space the fully discrete approximations is obtained with an implicit explicit method estimates for the continuous in time and the fully discrete methods are derived showing that the numerical approximation for the concentrations and the pressure are second order convergent in a discrete l 2 norm and in a discrete h 1 norm respectively

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

scaling of random fields by means of truncated power variograms and associated spectra an interpretation is offered for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with domain size we first demonstrate that the power semi variogram and associated spectra of random fields having homogeneous isotropic increments can be constructed as weighted integrals from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated homogeneous isotropic fields modes we then analyze the effect of filtering out truncating high and low frequency modes from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous with an autocovariance function that varies monotonically with separation distance in a manner not too dissimilar than that of its constituent modes the integral scales of the lowest and highest frequency modes cutoffs are related respectively to the length scales of the sampling window domain and data support sample volume taking each relationship to be one of proportionality renders our expressions for the integral scale and variance of a truncated field dependent on window and support scales in a manner consistent with observations the traditional approach of truncating power spectral densities yields autocovariance functions that oscillate about zero with finite in one and two dimensions or vanishing in one dimension integral scales our hierarchical theory allows bridging across scales at a specific locale by calibrating a truncated variogram model to data observed on a given support in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or larger one may also venture we suspect with less predictive power to bridge across both domain scales and locales by adopting generalized variogram parameters derived on the basis of juxtaposed hydraulic and tracer data from many sites

flow and travel time statistics in highly heterogeneous porous media 1 in this paper we present flow and travel time ensemble statistics based on a new simulation methodology the adaptive fup monte carlo method afmcm as a benchmark case we considered two dimensional steady flow in a rectangular domain characterized by multi gaussian heterogeneity structure with an isotropic exponential correlation and lnk variance y2 up to 8 advective transport is investigated using the travel time framework where lagrangian variables e g velocity transverse displacement or travel time depend on space rather than on time we find that eulerian and lagrangian velocity distributions diverge for increasing lnk variance due to enhanced channeling transverse displacement is a nonnormal for all y2 and control planes close to the injection area but after xiy 20 was found to be nearly normal even for high y2 travel time distribution deviates from the fickian model for large lnk variance and exhibits increasing skewness and a power law tail for large lnk variance the slope of which decreases for increasing distance from the source no anomalous features are found second moment of advective transport is analyzed with respect to the covariance of two lagrangian velocity variables slowness and slope which are directly related to the travel time and transverse displacement variance which are subsequently related to the longitudinal and transverse dispersion we provide simple estimators for the eulerian velocity variance travel time variance slowness and longitudinal dispersivity as a practical contribution of this analysis both two parameter models considered the advection dispersion equation and the lognormal model provide relatively poor representations of the initial part of the travel time probability density function in highly heterogeneous porous media we identify the need for further theoretical and experimental scrutiny of early arrival times and the need for computing higher order moments for a more accurate characterization of the travel time probability density function a brief discussion is presented on the challenges and extensions for which afmcm is suggested as a suitable approach

numerical simulation of non fickian transport in geological formations with multiple scale heterogeneities 1 we develop a numerical method to model contaminant transport in heterogeneous geological formations the method is based on a unified framework that takes into account the different levels of uncertainty often associated with characterizing heterogeneities at different spatial scales it treats the unresolved small scale heterogeneities residues probabilistically using a continuous time random walk ctrw formalism and the large scale heterogeneity variations trends deterministically this formulation leads to a fokker planck equation with a memory term fpme and a generalized concentration flux term the former term captures the non fickian behavior arising from the residues and the latter term accounts for the trends which are included with explicit treatment at the heterogeneity interfaces the memory term allows a transition between non fickian and fickian transport while the coupling of these dynamics with the trends quantifies the unique nature of the transport over a broad range of temporal and spatial scales the advection dispersion equation ade is shown to be a special case of our unified framework the solution of the ade is used as a reference for the fpme solutions for the same formation structure numerical treatment of the equations involves solution for the laplace transformed concentration by means of classical finite element methods and subsequent inversion in the time domain we use the numerical method to quantify transport in a two dimensional domain for different expressions for the memory term the parameters defining these expressions are measurable quantities the calculations demonstrate long tailing arising principally from the memory term and the effects on arrival times that are controlled largely by the generalized concentration flux term

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

pi degree parity in q skew polynomial rings abstract for k a field of arbitrary characteristic and r a k algebra we show that the pi degree of an iterated skew polynomial ring r x 1 1 1 x n n n agrees with the pi degree of r x 1 1 x n n when each i i satisfies a q i skew relation for q i k and extends to a higher q i skew i derivation we confirm the quantum gel fand kirillov conjecture for various quantized coordinate rings and calculate their pi degrees we extend these results to completely prime factor algebras

hermitian symmetric spaces and their unitary highest weight modules abstract the purpose of this article is to determine the set of unitarizable highest weight modules corresponding to hermitian symmetric spaces of the noncompact type the major step is that of proving unitarity at the last possible place with this established the description of the full set of unitarizable highest weight modules follows by a straightforward tensor product argument combined with the main ingredients of the proof of the key theorem bernstein gelfand gelfand and a diagramatic representation of the set of positive noncompact roots

concentration variance and spatial covariance in second order stationary heterogeneous conductivity fields the concentration variance and spatial covariance resulting from convective dispersive transport driven by a uniform mean flow in second order stationary conductivity fields was derived in a lagrangian framework based on a first order approximation of solute particle trajectories the approximation of the concentration co variances for large injection volumes is considerably simplified by defining the concentration at a certain location x and time t in terms of the backward solute trajectory probability distribution this is the probability that the trajectory of a fictitious microscopic and indivisible solute particle which is in a volume x centered around x at time t was at the time of solute injection t0 in the injection volume v0 the approximated concentration co variances were validated against concentration covariances derived from transport simulations in generated second order stationary conductivity fields the approximate solutions reproduced fairly well the effects of local scale dispersion and of the spatial variability of the hydraulic conductivity on the concentration co variance the effect of the spatial structure of the hydraulic conductivity field on the spatial covariance of the concentrations was investigated in order to identify parameters that can be unequivocally determined from the structure of the concentration field for a given spreading of the solute plume in the mean flow direction x11 t a given spatial correlation length of the loge transformed hydraulic conductivity in the transverse to flow direction i 2 and a given local scale dispersion dd the concentration covariance was nearly invariant and hardly influenced by the anisotropy of the covariance function e and the variance f2 of the loge transformed conductivity as a result e and f2 cannot be unequivocally determined from the spatial structure of the concentration field the concentration variance and spatial covariance in the transverse to mean flow direction are predominantly determined by the lateral component of the local scale dispersion dd22 and by the spatial correlation length of the loge transformed conductivity in the transverse to flow direction i 2 these two parameters might be unequivocally derived from the concentration co variance

contaminant spreading in stratified soils with fractal permeability distribution stochastic analysis of flow and transport in the subsurface usually assumes that the soil permeability is a stationary homogeneous stochastic process with a finite variance some field data suggest however that the permeability distributions may have a fractal character with long range correlations it is of interest to investigate how the fractal character of permeability distribution influences the spreading and mixing process in porous media the results of our analysis of this problem for perfectly stratified media with fractal distribution of permeability along the vertical are presented results were obtained for the transient and asymptotic dispersivities in the longitudinal direction the results show that the macroscopic asymptotic dispersivity depends strongly on the fractal dimension of vertical permeability distribution specifically the higher the fractal dimension the lower the value of macroscopic dispersivity the macroscopic dispersivity was found to be problem scale dependent in transient development and asymptotic phases variance of fluctuation of concentration was also analyzed and found to be dependent on the fractal dimension in this case the higher fractal dimension results in more mixing of pore water and therefore smoother smaller 2 concentration distribution

anisotropy lacunarity and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms it has been shown by di federico and neuman 1997 1998a b that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework which views hydraulic conductivity as a random fractal characterized by a power variogram such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments di federico and neuman 1997 have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated fields modes that are homogeneous and isotropic we show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic provided the ratios between principal integral spatial correlation scales are the same for all modes we then analyze the effect of filtering out truncating modes of low high and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous the integral scales of the lowest and highest frequency cutoff modes are related to length scales of the sampling window domain and data support sample volume respectively taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale in a manner previously shown to be consistent with observations in the isotropic case it also allows in principle bridging across scales at a specific locale as well as among locales by adopting either site specific or generalized variogram parameters the introduction of intermediate cutoffs allows us to account in a straightforward manner for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales for example where textural and structural features are associated with distinct ranges of scale such as fractures having discrete ranges of trace length and density which dissect the rock into matrix blocks having corresponding ranges of sizes we explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale variance covariance and spectra of a truncated fractal field we then develop an expression for the equivalent hydraulic conductivity of a box shaped porous block embedded within a multiscale log hydraulic conductivity field under mean uniform flow the block is larger than the support scale of the field but is smaller than a surrounding sampling window consequently its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function conditioned on a known mean value of support scale conductivity across the window are given explicitly by our multiscale theory

approximation of asymptotic dispersivity of conservative solute in unsaturated heterogeneous media with steady state flow the relation between the heterogeneity of hydraulic properties and effective asymptotic transport is studied for microscopically heterogeneous but macroscopically homogeneous unsaturated media with steady flow heterogeneity is described by the scaling of the hydraulic functions m and k the study is based on an analytical approximation of asymptotic dispersivity under the assumption that matric potential is spatially constant for the special case of a water saturated medium the result of the stochastic continuum theory is recovered when applied to several published numerical simulations the approximated asymptotic dispersivities are found to agree well with the numerical values by exploring the heterogeneous media for various hydraulic states the approximation resolves some apparent inconsistencies found in the simulations the cases considered span the entire range from perfect to zero correlation between scaling factors of matric potential and hydraulic conductivity it is demonstrated that this correlation is crucial for the behavior of asymptotic dispersivity with changing flow rate since weak to moderate correlations are often found in soils this result has significant implications for solute transport through heterogeneous soils

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

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

the significance of heterogeneity of evolving scales to transport in porous formations flow takes place in a heterogeneous formation of spatially variable conductivity which is modeled as a stationary space random function to model the variability at the regional scale the formation is viewed as one of a two dimensional horizontal structure a constant head gradient is applied on the formation boundary such that the flow is uniform in the mean a plume of inert solute is injected at t 0 in a volume v0 under ergodic conditions the plume centroid moves with the constant mean flow velocity u and a longitudinal macrodispersion coefficient dl may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid for a log conductivity covariance cy of finite integral scale i at first order in the variance y2 and for a travel distance l ut i dl y2ui and transport is coined as fickian ergodicity of the moments is ensured if l i where l is the initial plume scale some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with l without reaching a constant limit anomalous diffusion to model such a behavior previous studies have assumed that cy is stationary but of unbounded integral scale with cy ar 1 0 for large lag r under ergodic conditions it was found that asymptotically dl aul1 i e non fickian behavior and anomalous dispersion the present study claims that an ergodic behavior is not possible for a given finite plume of initial size l since the basic requirement that l i cannot be satisfied for cy of unbounded scale for instance the centroid does not move any more with u but is random figure 1 owing to the large scale heterogeneity in such a situation the actual effective dispersion coefficient dl is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization this is the accessible entity in a given experiment we show that in contrast with dl the behavior of dl is controlled by l and it has the fickian limit dl aul1 figure 3 we also discuss the case in which y is of stationary increments and is characterized by its variogram y then u and dl can be defined only if y is truncated equivalently an infrared cutoff is carried out in the spectrum of y however for a bounded u it is shown that dl depends only on y furthermore for y ar dl aul2l 1 i e dispersion is fickian for 0 1 whereas for 1 2 transport is non fickian since 2 dl cannot grow faster than l ut this is in contrast with a recently proposed model neuman 1990 in which the dispersion coefficient is independent of the plume size and it grows approximately like l1 5

a galerkin method for two dimensional hyperbolic integro differential equation with purely integral conditions the present paper is devoted to the solution for two dimensional hyperbolic integro differential equations subject to purely integral conditions first it is demonstrated the existence and uniqueness of the solution under certain conditions for the numerical approach a galerkin method based on least squares is proposed the numerical examples illustrate the technique and show the efficiency of the proposed method

variational theory and domain decomposition for nonlocal problems abstract in this article we present the first results on domain decomposition methods for nonlocal operators we present a nonlocal variational formulation for these operators and establish the well posedness of associated boundary value problems proving a nonlocal poincare inequality to determine the conditioning of the discretized operator we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix we then introduce a nonlocal two domain variational formulation utilizing nonlocal transmission conditions and prove equivalence with the single domain formulation a nonlocal schur complement is introduced we establish condition number bounds for the nonlocal stiffness and schur complement matrices supporting numerical experiments demonstrating the conditioning of the nonlocal one and two domain problems are presented

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

ensemble solute transport in two dimensional operator scaling random fields 1 motivated by field measurements of aquifer hydraulic conductivity k recent techniques were developed to construct anisotropic fractal random fields in which the scaling or self similarity parameter varies with direction and is defined by a matrix ensemble numerical results are analyzed for solute transport through these two dimensional operator scaling fractional brownian motion ln k fields both the longitudinal and transverse hurst coefficients as well as the radius of isotropy are important to both plume growth rates and the timing and duration of breakthrough it is possible to create operator scaling fractional brownian motion fields that have more continuity or stratification in the direction of transport the effects on a conservative solute plume are continually faster than fickian growth rates highly non gaussian shapes and a heavier tail early in the breakthrough curve contrary to some analytic stochastic theories for monofractal k fields the plume growth rates never exceed a mercado s 1967 purely stratified aquifer growth rate of plume apparent dispersivity proportional to mean distance apparent superstratified growth must be the result of other demonstrable factors such as initial plume size