on unifying the concepts of scale instrumentation and stochastics in the development of multiphase transport theory a generalized theory of multiphase transport is presented which combines the concepts of scale instrumentation stochastics and time series with the development of transport equations by defining the filtering process as a convolution of a measure p with a field property we are able to exploit the fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment an ideal instrument is defined which filters out high frequency noise corresponding to short distances and yet does not alter the structure of low frequencies using ideal instruments we successively filter out lower frequency noise in a multiscale multiphase environment formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process an equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed power spectra are developed which relate spectra on different scales to the measuring instrument used by successively applying filtering theorems a hierarchy of multiscale transport equations is developed filtered properties in the transport equations are mass averages different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases the concept of a wide sense stationary ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices

the goldie theorem for h semiprime algebras abstract the main result states that under certain assumptions about a hopf algebra h every h semiprime right noetherian h module algebra has a quasi frobenius classical right quotient ring another question treated in the paper is concerned with the extension of h module structures to quotient rings these results have an application to the semiprimeness problem for smash product algebras a h

solute dispersion in multidimensional periodic saturated porous media suppose in a convection dispersion equation governing solute movement in a saturated porous medium of infinite extent the convection velocity components are periodic functions of spatial coordinates then it follows from a general mathematical result that the solute concentration can be asymptotically approximated by a gaussian density two theoretical examples with and without a constant vertical velocity are given to illustrate an application of this mathematical result to solute dispersion in a parallel bedded three dimensional aquifer of infinite extent analytical expressions are obtained for dispersion coefficients in the asymptotic gaussian approximations in the two examples the dispersion coefficients functionally depend on a velocity scaling parameter u0 and a spatial scaling parameter period a via their product au0 and this functional dependence on au0 is different in the two examples the expressions for dispersion coefficients are formally contrasted and compared with those obtained previously by other authors in a parallel bedded infinite aquifer in which the convection velocity is a random field some physical implications of dependence of dispersion coefficients on a spatial scaling parameter i e the scale effect are discussed in the context of modeling in an aquifer with evolving heterogeneities this interpretation of scale effect in dispersion coefficients is contrasted with the preasymptotic interpretation in the current literature

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

nonerasing counting and majority over the linear time hierarchy in this paper we investigate several extensions of the linear time hierarchy denoted by lth we first prove that it is not necessary to erase the oracle tape between two successive oracle calls thereby lifting a common restriction on lth machines we also define a natural counting extension of lth and show that it corresponds to a robust notion of counting bounded arithmetic predicates finally we show that the computational power of the majority operator is equivalent to that of the exact counting operator in both contexts

enveloping algebras satisfying a polynomial identity in this paper we study ordinary and restricted enveloping algebras satisfying a polynomial identity we first show how the methods of j bergen and d s passman j algebra in press can be used to handle u l in all characteristics and u l if the latter ring is prime in particular we offer a simpler proof of the result of yu a bachturin j austral math soc 18 1974 10 21 on ordinary enveloping algebras in characteristic p 0 our main theorem asserts that in general u l is p i if and only if l has a restricted subalgebra of finite codimension which is essentially commutative these results are clearly the lie algebra analogs of the p i group ring theorems of m k smith j algebra 18 1971 477 499 and d s passman pacific j math 36 1971 467 483

semisimple hopf actions on weyl algebras abstract we study actions of semisimple hopf algebras h on weyl algebras a over an algebraically closed field of characteristic zero we show that the action of h on a must factor through a group action in other words if h acts inner faithfully on a then h is cocommutative the techniques used include reduction modulo a prime number and the study of semisimple cosemisimple hopf actions on division algebras

geologic heterogeneity representation using high order spatial cumulants for subsurface flow and transport simulations 1 the effects of geological heterogeneity representation on the hydraulic properties of two dimensional flow and transport simulations are studied using various stochastic simulation algorithms an alternative multiple point method hosim on the basis of high order spatial cumulants and legendre polynomials is used and compared to the multiple point filtersim method and the sequential gaussian simulation sgs method conditional realizations of a fluvial reservoir system are generated by hosim filtersim and sgs methods then the simulated hydraulic permeability fields k are used in a numerical groundwater flow and solute transport models numerical results showed that the hosim method created greater connectivity in the reservoir aquifer channel network than filtersim and sgs realizations the numerical simulations show that in a reservoir aquifer system with a strongly connected network of high k materials the gaussian and filtersim approaches are not as effective as hosim in reproducing this behavior the simulations showed a good agreement between hosim realizations and the exhaustive reference image

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

a full bayesian approach to parameter inference from tracer travel time moments and investigation of scale effects at the cape cod experimental site a method for inverting the travel time moments of solutes in heterogeneous aquifers is presented and is based on peak concentration arrival times as measured at various samplers in an aquifer the approach combines a lagrangian rubin and dagan 1992 solute transport framework with full bayesian hydrogeological parameter inference in the full bayesian approach the noise values in the observed data are treated as hyperparameters and their effects are removed by marginalization the prior probability density functions pdfs for the model parameters horizontal integral scale velocity and log k variance and noise values are represented by prior pdfs developed from minimum relative entropy considerations analysis of the cape cod massachusetts field experiment is presented inverse results for the hydraulic parameters indicate an expected value for the velocity variance of log hydraulic conductivity and horizontal integral scale of 0 42 m d 0 26 and 3 0 m respectively while these results are consistent with various direct field determinations the importance of the findings is in the reduction of confidence range about the various expected values on selected control planes we compare observed travel time frequency histograms with the theoretical pdf conditioned on the observed travel time moments we observe a positive skew in the travel time pdf which tends to decrease as the travel time distance grows we also test the hypothesis that there is no scale dependence of the integral scale with the scale of the experiment at cape cod we adopt two strategies the first strategy is to use subsets of the full data set and then to see if the resulting parameter fits are different as we use different data from control planes at expanding distances from the source the second approach is from the viewpoint of entropy concentration no increase in integral scale with distance is inferred from either approach over the range of the cape cod tracer experiment

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

a pi degree theorem for quantum deformations abstract we prove that if a filtered quantization a of a finitely generated commutative domain over a field k is a pi algebra then a is commutative if char k 0 and its pi degree is a power of p if char k p

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

arithmetical definability and computational complexity in this paper we introduce and study some syntactical fragments of monadic second order and first order peano arithmetic which we will prove the connection to famous complexity classes starting from descriptive complexity results and giving an effective method for translating formulas between different logical structures representing encodings of integers we give some new arithmetical characterizations of np ph nl and p

finite dimensional hopf actions on weyl algebras abstract we prove that any action of a finite dimensional hopf algebra h on a weyl algebra a over an algebraically closed field of characteristic zero factors through a group action in other words weyl algebras do not admit genuine finite quantum symmetries this improves a previous result by the authors where the statement was established for semisimple h the proof relies on a refinement of the method previously used namely considering reductions of the action of h on a modulo prime powers rather than primes we also show that the result holds more generally for algebras of differential operators this gives an affirmative answer to a question posed by the last two authors

a quasi linear theory of non fickian and fickian subsurface dispersion 1 theoretical analysis with application to isotropic media a theory is presented which accounts for nonlinearity caused by the deviation of plume particles from their mean trajectory in three dimensional statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities existing linear theories predict that in the absence of local dispersion transverse dispersivities tend asymptotically to zero as fickian conditions are reached according to our new quasi linear theory these dispersivities ascend to peak values and then diminish gradually toward nonzero fickian asymptotes which are proportional to y4 when the log hydraulic conductivity variance y2 is much less than 1 all existing theories agree that in isotropic media the asymptotic longitudinal dispersivity is proportional to y2 when y2 1 and all are nominally restricted to mildly heterogeneous media in which this inequality is satisfied however the quasi linear theory appears to be less prone to error than linear theories when extended to strongly heterogeneous media because it deals with the above nonlinearity without formally limiting y2 it predicts that when y 1 in isotropic media both the longitudinal and transverse dispersivities ascend monotonically toward fickian asymptotes proportional to y

extensions of mso and the monadic counting hierarchy in this paper we study the expressive power of the extension of first order logic by the unary second order majority quantifier most1 in 1 it was shown that the extension of fo by second order majority quantifiers of all arities describes exactly the problems in the counting hierarchy we consider first certain sublogics of fo most1 over unary vocabularies we show that over unary vocabularies the logic mso r where mso is monadic second order logic and r is the first order rescher quantifier can be characterized by presburger arithmetic whereas the logic mso rn n z where rn is the nth vectorization of r corresponds to the 0 fragment of arithmetic then we show that fo most1 mso rn n z and that on unary vocabularies fo most1 collapses to uniform tc0 using this collapse we show that first order logic with the binary second order majority quantifier is strictly more expressive than fo most1 over the empty vocabulary on the other hand over strings fo most1 is shown to capture the linear fragment of the counting hierarchy finally we show that over non unary vocabularies fo most1 can express problems complete via first order reductions for each level of the counting hierarchy

extensions of valuation rings in central simple algebras certain subrings r of simple algebras q finite dimensional over their center k are studied these rings are called q valuation rings since they share many properties with commutative valuation rings let v be a valuation ring of k the center of q and let m be the set of q valuation rings r in q with r n k v then 1s 1 this extension theorem which does not hold if one considers only total valuation rings was proved by n i dubrovin here first a somewhat different proof of this result is given and then information about the set m is obtained theorem the elements in j are conjugate if v has finite rank theorem the elements in m are total valuation rings if m contains one total valuation ring in this case q is a division ring theorem i i 1 if 5 contains an invariant total valuation ring

rudimentary relations and primitive recursion a toolbox abstract rudimentary relations are those relations over natural integers that are defined by a first order arithmetical formula in which all quantifications are bounded by some variables the question of whether a given primitive recursive relation is rudimentary is in some cases difficult and related to several well known open questions in theoretical computer science in this paper we present systematic tools to study this question and various applications one of them gives a sufficient condition of the collapsing of the first classes of the grzegorczyk s hierarchy

matrix rings and linear groups over a field of fractions of enveloping algebras and group rings i 1 1 let h be a lie algebra over a field k u h be its universal envelope it is well known that u h is a domain see 1 section v 31 and the theorem of cohn see 2 states that u h can be imbedded in a field when h is locally finite u h has even a field of fractions see 1 theorem v 3 61 the class of lie algebras whose universal envelope has a field of fractions is however much wider than the class of locally finite lie algebras it follows from proposition 4 1 of the article that it contains for instance all the locally soluble by locally finite algebras this gives a negative answer of the question in 2 page 528 let h satisfy any one of the following three conditions

eulerian lagrangian analysis of transport conditioned on hydraulic data 2 effects of log transmissivity and hydraulic head measurements in paper 1 of this series we described an analytical numerical method to predict deterministically solute transport under uncertainty the method is based on a unified eulerian lagrangian theory which allows conditioning of the predictions on hydraulic measurements conditioning on measured concentrations is also possible as demonstrated by neuman et al 1993 in this paper we condition velocity on log transmissivity and or hydraulic head data via cokriging we then combine an early time analytical solution with a pseudo fickian galerkin finite element scheme for later time to obtain conditional predictions of concentration and lower bounds on its variance and coefficient of variation the pseudo fickian scheme involves a conditional dispersion tensor which depends on information and varies in space time hence the predicted plumes travel along curved trajectories and attain irregular non gaussian shapes we also compute a measure of uncertainty for the original source location of a solute particle whose position at some later time is known from sampling spatial maps of this particle origin covariance provide vivid images of preferential flow paths and exclusion zones identified by the available data we illustrate these concepts and results on instantaneous point and area sources

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

on the existence of orders in semisimple hopf algebras the first author was supported by the projects mtm2011 27090 from micinn and feder and by the research group fqm0211 from junta de andalucia the second author was supported by the danish national research foundation dnrf through the centre for symmetry and deformation the authors would like to thank dmitriy rumynin and martin lorenz for bring ing to their attention the notion of weak order cesar galindo for pointing out that these examples were already discussed in 8 and yevgenia kashina and sonia na tale for useful discussions on the classification of low dimensional semisimple hopf algebras

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

semisimple hopf actions on commutative domains abstract let h be a semisimple so finite dimensional hopf algebra over an algebraically closed field k of characteristic zero and let a be a commutative domain over k we show that if a arises as an h module algebra via an inner faithful h action then h must be a group algebra this answers a question of e kirkman and j kuzmanovich and partially answers a question of m cohen the main results of this article extend to working over k of positive characteristic on the other hand we obtain results on hopf actions on weyl algebras as a consequence of the main theorem