families of minimal involutive surfaces in projective space abstract we present an algorithm that can be used to determine if a given surface in complex projective 3 space is minimal involutive and we apply it to the construction of explicit examples of families of minimal involutive surfaces of degrees 3 and 4

steady state groundwater flow simulation with imprecise parameters a methodology based on fuzzy set theory is developed to incorporate imprecise parameters into steady state groundwater flow models in this case fuzzy numbers are used to represent parameter imprecision as such they are also used as a measure for the uncertainty associated with the hydraulic heads due to the imprecision in the input parameters the imprecise input parameters may come from indirect measurements subjective interpretation and expert judgment of available information in the methodology a finite difference method is combined with level set operations to formulate the fuzzy groundwater flow model this fuzzy modeling technique can handle imprecise parameters in a direct way without generating a large number of realizations two numerical solution methods are used to solve the fuzzy groundwater flow model the groundwater model operator method proposed in this methodology and the iterative algorithm based on conventional interval arithmetics the iterative method is simple but may overestimate the uncertainty of hydraulic heads the groundwater model operator method not only provides the hull of the solution set for the hydraulic heads but also considers the dependence of hydraulic head coefficients which are functions of imprecise parameters sensitivity analysis shows that the dependence of hydraulic head coefficients has a critical impact on the model results and neglecting this dependence may result in significant overestimation of the uncertainty of hydraulic heads a numerical model based on the methodology is tested by comparing it with the analytical solution for a homogeneous radial flow problem it is also applied to a simplified two dimensional heterogeneous flow case to demonstrate the methodology

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

krull dimension of generalized weyl algebras with noncommutative coefficients abstract we obtain formulae for the krull dimension of the generalized weyl algebra t r a where r is a left noetherian ring

on the estimation of pareto fronts from the point of view of copula theory a probabilistic framework is applied to pareto front estimation a link between pareto front and level lines of a multivariate distribution is given parametric approximations of pareto front are provided for archimedean copulas consequences and limits of the archimedean hypothesis are discussed the full estimation methodology is analyzed on several examples given a first set of observations from a design of experiments sampled randomly in the design space the corresponding set of non dominated points usually does not give a good approximation of the pareto front we propose here to study this problem from the point of view of multivariate analysis introducing a probabilistic framework with the use of copulas this approach enables the expression of level lines in the objective space giving an estimation of the position of the pareto front when the level tends to zero in particular when it is possible to use archimedean copulas analytical expressions for pareto front estimators are available several case studies illustrate the interest of the approach which can be used at the beginning of the optimization when sampling randomly in the design space

characteristic varieties and vanishing cycles on presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres ce sont des sous varietes lagrangiennes coniques dans le fibre cotangent t x

nonholonomic modules over enveloping algebras of semisimple lie algebras abstract we prove that the enveloping algebras of all semisimple lie algebras over the complex numbers which are not of type a 1 admit nonholonomic simple modules

stochastic models of subsurface flow log linearized gaussian models are exact for covariances stochastic solutions for steady state infinite domain flow models are studied under the assumptions that log conductivity and head are second order stationary jointly gaussian processes it is shown that the covariance relationships previously derived are exact solutions for the nonlinearized equations no matter how large the input variance is

an application of the geostatistical approach to the inverse problem in two dimensional groundwater modeling the geostatistical approach to the estimation of transmissivity from head and transmissivity measurements is developed for two dimensional steady flow the field of the logarithm of transmissivity log transmissivity is represented as a zero order intrinsic random field its spatial structure is described in this application through a two term covariance function that is linear in the parameters 1 and 2 linearization of the discretized flow equations allows the construction of the joint covariance matrix of the head and log transmissivity measurements as a linear function of 1 and 2 in this particular application the coefficient matrices are calculated numerically in a noniterative fashion maximum likelihood estimation is employed to estimate 1 and 2 as well as additional parameters from measurements linear estimation theory cokriging then yields point or block averaged estimates of transmissivity the approach is first applied to a test case with favorable results it is shown that the application of the methodology gives good estimates of transmissivities it is also shown that when the transmissivities are used in a numerical model they reproduce the head measurements quite well results from the application of the methodology to the jordan aquifer in iowa are also presented

dixmier s problem 5 for the weyl algebra abstract the purpose of this paper is to give a negative answer to problem 5 and a positive answer in the case of homogeneous elements to problem 4 of the article of j dixmier bull soc math france 96 1968 209 242 on the first weyl algebra

foliations of multiprojective spaces and a conjecture of bernstein and lunts we use foliations of multiprojective spaces defined by hamiltonian functions on the underlying affine space to prove the three dimensional case of a conjecture of bernstein and lunts according to which the symbol of a generic first order differential operator gives rise to a hypersurface of the cotangent bundle which does not contain involutive conical subvarieties apart from the zero section and fibres of the bundle

extensions of modules over weyl algebras in this paper we calculate some ext groups of singular modules over the complex weyl algebra an in particular we determine conditions under which ext is an infinite dimensional vector space when n 2 or 3

the dixmier conjecture and the shape of possible counterexamples we establish a lower bound for the size of possible counterexamples of the dixmier conjecture we prove that b 15 where b is the minimum of the greatest common divisor of the total degrees of p and q where p q runs over the counterexamples of the dixmier conjecture 2013 elsevier inc

simple holonomic modules over the second weyl algebra a2 abstract for simple generalized weyl algebras of gelfand kirillov dimension 4 a class including the second weyl algebra a 2 some simple factor algebras of the universal enveloping algebra of the lie algebra sl 2 sl 2 and of u q sl 2 u q sl 2 etc the simple holonomic modules are classified up to pairs of irreducible elements of certain noncommutative euclidean ring

stochastic analysis of spatial variability in two dimensional steady groundwater flow assuming stationary and nonstationary heads two dimensional steady groundwater flow in a confined aquifer with spatially variable transmissivity t is analyzed stochastically using spectral analysis and the theory of intrinsic random functions conditions that ensure a stationary statistically homogeneous head process are derived and using two convenient forms for the covariance function of the ln t process the head covariance function is studied in addition the head variogram is obtained for a particular nonstationary case and the asymptotic head variogram is derived under very general conditions results are compared to those obtained by gelhar 1976 for one and two dimensional phreatic flow and bakr et al 1978 for one and three dimensional confined flow multidimensional flow analysis results in a significantly reduced head variance the head correlation remains high over much greater distances than the ln t correlation the variogram obtained when stationary heads are assumed is identical to that obtained for nonstationary heads for dimensionless lag distances up to 2 times the correlation scale of the log transmissivity the variogram for nonstationary heads continues to grow logarithmically as lag distance increases independent of the form of the input covariance in the nonstationary case the conditions for stationarity are contrasted with the corresponding results obtained for the one and three dimensional cases of gutjahr and gelhar 1981 the head variance calculated from the stationary theory is found to agree with that of previous monte carlo simulations

analysis of flow through heterogeneous random aquifers 2 unsteady flow in confined formations the direct problem of determining the head field in a formation of given stationary isotropic random structure is solved approximately for unsteady flow conditions under restrictive assumptions of sufficiently small conductivity variance and average head gradient slowly varying in time and space closed form expressions are derived for the effective conductivity or storativity and for the head variogram two types of unsteady flow are considered transients and periodic flows in the case of transients it is assumed that initially the head is constant no flow and that ultimately the flow becomes uniform it is found that the effective conductivity and transmissivity are time dependent and drop from the arithmetic mean initially to the steady state value during a relaxation time the latter is much larger for one dimensional flows than for two or three dimensional ones the head variogram also relaxes from the one corresponding to the initial lack of correlation to the steady state variogram during a relaxation time which grows with the lag while in three dimensional structure the relaxation time scales are relatively short and steady state analysis can be used accurately even under transient conditions the tendency to steady state may be very slow in the case of transmissivity because of its large integral scale hence steady state analysis for transients might not be warranted for regional transient flows in the case of periodic flows of seasonal nature the head gradient is assumed to be made up from a steady component and a periodic one if the amplitude of the latter is sufficiently small compared to the steady one steady state value of effective transmissivity and storativity can be adopted the head variogram has also its steady state form for large lags and quasisteady for small lags hence the variogram lies between the steady one steady component of head gradient and the quasisteady one instantaneous head gradient

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

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

an analogue of the conjecture of dixmier is true for the algebra of polynomial integro differential operators abstract let a 1 k x d d x be the weyl algebra and i 1 k x d d x be the algebra of polynomial integro differential operators over a field k of characteristic zero the conjecture problem of dixmier 1968 still open is an algebra endomorphism of the weyl algebra a 1 an automorphism the aim of the paper is to prove that each algebra endomorphism of the algebra i 1 is an automorphism notice that in contrast to the weyl algebra a 1 the algebra i 1 is a non simple non noetherian algebra which is not a domain moreover it contains infinite direct sums of nonzero left and right ideals

on involutive homogeneous varieties and representations of weyl algebras abstract this paper is concerned with the geometry of minimal involutive homogeneous varieties in complex affine 2 n space and its application to the study of the representation theory of the n th complex weyl algebra a n the main results are the existence of minimal involutive homogeneous varieties of any given codimension and of a n modules that have these varieties for characteristic variety we also determine conditions on the codimensions of a n modules m and n under which ext 1 m n is a finite dimensional vector space

a generic cubic surface contains no involutive curve abstract in 1988 j bernstein and v lunts invent math 94 223 243 proved that a generic surface in p 3 of degree 4 or greater would not contain a curve which supported a module over the second weyl algebra it is known that quadric surfaces and planes always do this paper shows that a generic cubic surface does not

stochastic modeling of groundwater flow by unconditional and conditional probabilities 1 conditional simulation and the direct problem mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated the formation properties hydraulic conductivity transmissivity as well as flow variables head specific discharge solute concentration are regarded as random variables subjected to uncertainty the main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables the unconditional probability density functions are supposed to be stationary multivariate normal while conditioning accounts for the measured values at a few points of the formation two problems of groundwater flow are discussed in part 1 conditional simulation and the direct problem for steady flow analytical results are obtained by using perturbation approximations average uniform head gradient flows as well as recharge and flow to wells are discussed in unconditional modeling the input variable the conductivity or transmissivity log is regarded as stationary and is represented by its constant mean and variogram the ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions in conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points consequently statistical computations are performed for the subensemble of aquifers which preserve the measured values and as a result both input and output variables are nonstationary the main effect of conditioning is to reduce the variance i e the uncertainty of these variables this effect is particularly important in the case of flow toward wells application of the modeling method to field problems is outlined

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

a note on higher order corrections of the head covariances in steady aquifer flow average uniform flow takes place in a heterogeneous aquifer of infinite extent the input to the problem is the hydraulic conductivity which is regarded as a random space function that is lognormal stationary and statistically isotropic the output is the water head field which is a random function satisfying the equation of steady flow the first order approximation of the head in an asymptotic expansion for small log conductivity variance is a normal function characterized completely by the head log conductivity cross covariance and the head covariance or variogram these covariances are proportional to the log conductivity variance by using spectral methods second order corrections of the head covariances proportional to the log conductivity variance squared are derived explicitly detailed calculations are carried out for an exponential log conductivity covariance the main finding of the note is that the first order approximation is very robust and even for a log conductivity variance equal to unity the second order correction of the head variances is smaller than 10 of the first order approximation

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

stochastic models of subsurface flow infinite versus finite domains and stationarity stochastic solutions of the differential equation describing one dimensional flow through a porous medium with spatially variable hydraulic conductivity which is represented by a stationary statistically homogeneous random process are developed using several techniques the analytical approximations using first order analysis propagation of error covariance differential equations and variogram analysis all yield consistent results which demonstrate the important effects of boundary conditions and conditioning using the theory of intrinsic random functions stochastic solutions are developed for the case when the logarithm of the hydraulic conductivity is a three dimensional stationary random field in contrast to the one dimensional case it is found that the resulting three dimensional head perturbation will be locally stationary under very general conditions results from the one dimensional analytical solutions are found to be in agreement with previous monte carlo simulations for a flow system of finite length the solution based on linearization in the logarithm of hydraulic conductivity proved to be very robust showing reasonable agreement with monte carlo results even for the largest input standard deviation of lnk 4 6