moment maps and riemannian symmetric pairs we study hamiltonian actions of a compact lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold the fixed point set of the involution on the manifold is a lagrangian submanifold we investigate its image under the moment map and conclude that the intersection with the weyl chamber is an easily described subpolytope of the kirwan polytope of special interest is the integral kahler case where much stronger results hold in particular we obtain convexity theorems for closures of orbits of the noncompact dual group in the sense of the theory of symmetric pairs in the abelian case these results were obtained earlier by duistermaat we derive explicit inequalities for polytopes associated with real flag varieties

transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations in a recent paper we developed exact nonlocal conditional moment equations for transient flow in bounded domains driven by random forcing terms sources initial head and boundary conditions whereas our conditional mean equations took into account the randomness of forcing terms our conditional second moment equations did not we do so in this brief addendum

solutions of a system of functional differential equations in a right neighborhood of zero abstract we give some sufficient conditions for the system of nonlinear functional differential equations x qt x t f t x t x 1 qt x 2 qt x k qt x qt x g t where q 0 1 and functions f 0 t r n r n r n r n j 0 t r n 0 t j 1 k and g 0 t 0 t are continuous for the unique existence of continuously differentiable solutions x t on the interval 0 t 0 t satisfying the condition x 0 0

the geostatistical characteristics of the borden aquifer a complete reexamination of sudicky s 1986 field experiment for the geostatistical characterization of hydraulic conductivity at the borden aquifer in ontario canada is performed the sampled data reveal that a number of outliers low ln k values are present in the data base these low values cause difficulties in both variogram estimation and determining population statistics the analysis shows that assuming either a normal distribution or exponential distribution for log conductivity is appropriate the classical cressie hawkins and squared median of the absolute deviations smad estimators are used to compute experimental variograms none of these estimators provides completely satisfactory variograms for the borden data with the exception of the classical estimator with outliers removed from the data set theoretical exponential variogram parameters are determined from nonlinear nl estimation differences are obtained between nl fits and those of sudicky 1986 for the classical screened estimated variogram nl fits produce an ln k variance of 0 24 nugget of 0 07 and integral scales of 5 1 m horizontal and 0 21 m vertical along a a for b b these values are 0 37 0 11 8 3 and 0 34 the fitted parameter set for b b data horizontal and vertical is statistically different than the parameter set determined for a a we also evaluate a probabilistic form of dagan s 1982 1987 equations relating geostatistical parameters to a tracer cloud s spreading moments the equations are evaluated using the parameter estimates and covariances determined from line a a as input with a velocity equal to 9 0 cm day the results are compared with actual values determined from the field test but evaluated by both freyberg 1986 and rajaram and gelhar 1988 the geostatistical parameters developed from this study produce an excellent fit to both sets of calculated plume moments when combined with dagan s stochastic theory for predicting the spread of a tracer cloud

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

three dimensional steady state flow to a well in a randomly heterogeneous bounded aquifer 1 we consider flow in a confined aquifer of uniform thickness due to a well of zero radius that fully penetrates the aquifer and discharges at a constant rate if the lateral extent of the aquifer is infinite a steady state flow regime never develops it is however well known that if the aquifer is additionally uniform a quasi steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time on the expanding surface head is uniform and time invariant inside this surface head at any time is described by a steady state solution a rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three dimensional transient stochastic flow problem in an aquifer of infinite lateral extent here we take a different approach by developing a three dimensional steady solution for mean flow to a well in a randomly heterogeneous aquifer with a cylindrical prescribed head boundary in analogy to the uniform case we expect our solution to approximate a quasi steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger we treat log conductivity as a statistically homogeneous random field characterized by a gaussian spatial covariance function that may have different horizontal and vertical correlation scales our solution consists of analytical expressions for the ensemble mean and variance of head in the aquifer to second order in the standard deviation of log conductivity it is based on recursive approximations of exact nonlocal moment equations that are free of distributional assumptions and so apply to both gaussian and non gaussian log conductivity fields the analytical solution is supported by numerical monte carlo simulations it clarifies the manner in which relationships between the horizontal and vertical scales of the quasi steady state region and those of statistical anisotropy impact the statistical moments of drawdown and the equivalent and apparent hydraulic conductivities of the aquifer both conductivities are shown to exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal integral scales from it

flow to partially penetrating wells in unconfined heterogeneous aquifers mean head and interpretation of pumping tests 1 a partially penetrating well of length lw and radius rw starts to pump at constant discharge qw at t 0 from an unconfined aquifer of thickness d the aquifer is of random and stationary conductivity characterized by kg geometric mean y2 log conductivity variance and i and iv the horizontal and vertical integral scales the flow problem is solved under a few simplifying assumptions commonly adopted in the literature for homogeneous media rw lw 1 linearization of the free surface condition and constant drainable porosity n additionally it is assumed that rw i 1 and lw iv 1 to simplify the well boundary conditions and that a first order approximation in y2 extended to finite y2 on a conjectural basis is adopted the solution is obtained for the mean head field h r z t and the associated water table equation the main result of the analysis is that the flow domain can be divided into three zones for h 1 the neighborhood of the well r i where h qw lwka h0 r z tkefuv nd with h0 being the zero order solution pertaining to a homogeneous and isotropic aquifer ka being the conductivity arithmetic mean and kefuv being the effective vertical conductivity in mean uniform flow 2 an exterior zone r i in which h qw lwkefuh h0 r z tkefuv nd with kefuh being the horizontal effective conductivity and 3 an intermediate zone in which the solution requires a few numerical quadratures not carried out here the application to pumping tests reveals that identification of the aquifer parameters for homogeneous and anisotropic aquifers by commonly used methods can be applied for the drawdown measured in an observation well of length lowiv to ensure exchange of space and ensemble head averages in the second zone in order to identify kefuh kefuv and n in contrast the use of the drawdown in the well first zone leads to an overestimation of kefuh by the factor ka kefuh

on periodic and solutions converging to zero of some systems of differential difference equations sufficient conditions for the existence of a periodic c 1 solution of a system of differential difference equations are given we also give some sufficient conditions for the existence of an n parameter family of solutions of a system of differential difference equations converging to zero along with their first derivatives

the extended thiem s solution including the impact of heterogeneity 1 in this study we present a formula for the hydraulic head describing the mean drawdown of a three dimensional steady state pumping test in heterogeneous anisotropic porous media effectively by modeling the hydraulic conductivity k x as spatial random function and using the upscaling method coarse graining we succeed in deriving a closed form solution hefw r which we understand as an extension of thiem s formula to heterogeneous media the solution hefw r does not only depend on the radial distance r but accounts also for the statistics of k x namely geometric mean kg variance 2 horizontal correlation length l and anisotropy ratio e we perform a sensitivity analysis on the parameters of hefw r and implement an inverse estimation strategy using numerical pumping tests we show the applicability of hefw r on the interpretation of drawdown data this will be done for both an ensemble of as well as for single pumping tests making use of the inverse estimation method we find excellent agreement of estimated parameters with initial values in particular for the horizontal correlation length

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

on some solvable systems of difference equations abstract we show that the following systems of difference equations x n 1 u n 1 v n y n 1 w n 1 s n n n 0 where u n v n w n s n are some of the sequences x n or y n with real initial values x 0 and y 0 are solvable in fourteen out of sixteen possible cases two cases are left unsolved probably the most interesting is the result in the case u n x n v n x n w n x n s n y n where a fascinating formula is obtained in an elegant way by using some ad hoc ideas

uncertainty quantification in scale dependent models of flow in porous media equations governing flow and transport in randomly heterogeneous porous media are stochastic and scale dependent in the moment equations me method exact deterministic equations for the leading moments of state variables are obtained at the same support scale as the governing equations computable approximations of the mes can be derived via perturbation expansion in orders of the standard deviation of the random model parameters as such their convergence is guaranteed only for standard deviation smaller than one here we consider steady state saturated flow in a porous medium with random second order stationary conductivity field we show it is possible to identify a support scale where the typically employed approximate formulations of me yield accurate statistical moments of a target state variable therefore at support scale and larger me presents an attractive alternative to slowly convergent monte carlo mc methods whenever lead order statistical moments of a target state variable are needed we also demonstrate that a surrogate model for statistical moments could be constructed from mc simulations at larger support scales and be used to accurately estimate moments at smaller scales where mc simulations are expensive and the me method is not applicable

radially convergent flow in heterogeneous porous media we present an analytical solution for apparent effective transmissivity under radially convergent steady state flow conditions produced by constant pumping from a single well of finite radius rw apparent effective transmissivity te is defined as the value that relates the expected values of flow and head gradient at a certain location the domain is two dimensional of annular shape and the size of the pumping well is explicitly taken into account the solution for the steady state heads is obtained by solving the perturbed flow equation and substituting it into darcy s law to obtain a consistent second order expansion for te we show that apparent effective transmissivity is a scalar for any choice of isotropic covariance model with an expression given in integral form our main result is that te in a heterogeneous statistically isotropic random field under radial steady state flow conditions is a monotonie increasing function of r distance from the well that rises from the harmonic mean of the point transmissivity values close to the well and tends asymptotically towards the geometric mean far from the well the asymptotic value is reached at a distance of a few integral scales 1 5 2 for the gaussian model and 3 5 for the exponential one the apparent effective transmissivity versus normalized r curves are in excellent agreement with previously published numerical work carried out using monte carlo method

a note on well boundary condition for flow through heterogeneous formations 1 in modeling flow toward wells it is mathematically convenient to replace the well by a singularity line along its axis in the case of homogeneous aquifers confined flow and fully penetrating wells the strength of the sources is constant and the common condition of constant head on the well is satisfied exactly it also leads to constant flux on the well envelope in the case of aquifers of spatially variable conductivity the constant head boundary condition can be satisfied accurately if the well radius is much smaller than the log conductivity horizontal integral scale by selecting a variable source strength that is proportional to the local conductivity along the well indelman et al 1996 fiori et al 1998 indelman 2003a 2003b in this case the equivalent conductivity tends to the arithmetic mean near the well it changes with distance approaching from above the effective conductivity in uniform horizontal mean flow far from the well selecting inadvertently a constant source strength violates the condition of constant well head the equivalent conductivity tends to the harmonic mean near the well and approaches from below the effective one in uniform mean flow the equivalent conductivity is derived for both types of sources for stratified formations as well as for aquifers of a three dimensional heterogeneous structure the implications on interpretation of head measurements by the two models is illustrated by a few examples

retract principle for neutral functional differential equations abstract the investigation of asymptotic behaviour of solutions of ordinary differential equations is often based on the application of the retract principle initially developed for ordinary differential equations this technique was extended to other classes of equations not answered remains a problem concerning the possibility of extending this principle to neutral differential equations the goal of the present paper is to partially fill this gap and develop a corresponding technique for the application of this principle the applicability of the main result is illustrated on a nonlinear equation and sufficient conditions for existence of a positive solution are derived

asymptotically convergent solutions of a system of nonlinear functional differential equations of neutral type with iterated deviating arguments some conditions that guarantee the existence of a family of solutions converging to zero of a system of nonlinear functional differential equations with iterated deviating arguments are given

unique existence of bounded continuous solutions on the real line of a class of nonlinear functional equations with complicated deviations abstract the existence and uniqueness of bounded continuous solutions on the real line of a general class of nonlinear functional equations with quite complicated deviations dependent of unknown functions are established in this paper

on solutions of a class of systems of nonlinear functional equations in a neighborhood of zero satisfying lipschitz condition unique existence of solutions of a system of nonlinear functional equations with iterated deviations in a neighborhood of zero and satisfying the lipschitz condition is studied here

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 2 computational examples in paper 1 guadagnini and neuman this issue of this two part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded randomly nonuniform domains in the presence of uncertain source and boundary terms we derived recursive approximations for these equations based on expansion in powers of a small parameter y which represents the standard estimation error of log hydraulic conductivity developed a two dimensional finite element scheme for their solution under the action of deterministic forcing terms to first order in y2 showed how to localize the conditional mean flow equations so as to yield familiar looking darcian differential equations in which however all quantities are conditional on data and therefore nonunique and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations we stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling from support to grid scale or the a priori definition of a computational grid that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for monte carlo simulations this notwithstanding we use in this paper a high resolution grid to explore superimposed mean uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal localized and monte carlo finite element solutions whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 we find that it actually yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 4 localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as y2 increases though they require much less computer time than do nonlocal solutions localized solutions have the added disadvantage that they yield no information about predictive uncertainty

effective groundwater model parameter values influence of spatial variability of hydraulic conductivity leakance and recharge a stochastic simulation approach was used to inspect the influence of spatial variability in aquifer and recharge properties on effective averaged groundwater model parameters a two dimensional unconfined aquifer model was set up for an area similar to that near livermore california monte carlo simulations were generated for different sets of spatial correlation structures assuming stationarity and an exponential semivariogram for each monte carlo realization groundwater flow was simulated and estimates of the means and standard deviations of the hydraulic head were obtained nine different cases were studied involving different correlation lengths of hydraulic conductivity conditional simulations parameter zonation and spatial recharge distributions results indicate that there is no single uniform value of the hydraulic conductivity that reproduces the expected head values spatial averaging to obtain effective values was based on the p norm power norm which ranges from 1 for the arithmetic mean to 1 for the harmonic mean with 0 representing the geometric mean for the unconditional case the 0 4 p norm seems to best reproduce the expected values the 0 2 p norm gave the best result for the conditional simulation case if the geometric mean were used instead the heads would deviate from the expected heads by about 2 m near the wells some value between the arithmetic and the geometric means of riverbed leakance will give results close to the expected head values the arithmetic mean of aerially distributed recharge produced results with small deviations from the expected head values these particular results may only apply to this modeled system as the pumping pattern and magnitudes exhibited strong influences on zones of maximum head deviation the implication of this work is that if an effective hydraulic conductivity is used in a simulation model it must be selected for a particular set of wells and pumping rates in an aquifer with significant pumping centers the best effective value for a two dimensional unconfined flow model is most likely between the geometric and harmonic mean if the wells are turned off the effective hydraulic conductivity for that flow model reverts to the geometric mean

an evaluation of jacob s method for the interpretation of pumping tests in heterogeneous formations most pumping tests are interpreted using the classical theis assumption of large scale homogeneity with various corrections to account for early time behavior of drawdown curves when drawdowns are plotted versus log time late time data often delineate a straight line which is consistent with jacob s approximation of theis solution but may seem surprising in view of the heterogeneity of natural media the aim of our work is to show that jacob s method leads to a good approximation of the effective transmissivity of heterogeneous media when constrained to late time data a review of several multiwell pumping tests demonstrates that when drawdown curves from each observation well are interpreted separately they produce very similar transmissivity t estimates however the corresponding estimates for storativity span a broad range this behavior is verified numerically for several models of formation heterogeneity a very significant finding of the numerical investigation is that t values estimated using simulated drawdown from individual observation wells are all very close to the effective t value for parallel flow this was observed even in nonmultigaussian t fields where high t zones are well connected and where the effective t is larger than the geometric average of point values this implies that jacob s method can be used for estimating effective t values in many if not most formations

two dimensional characterization of hydraulic heterogeneity by multiple pumping tests 1 the conventional analysis of pumping tests by type curve methods is based on the assumption of a homogeneous aquifer applying these techniques to pumping test data from real heterogeneous aquifers leads to estimates of the hydraulic parameters that depend on the choice of the pumping and observation well positions in this paper we test whether these values may be viewed as pseudo local values of transmissivity and storativity which can be interpolated by kriging we compare such estimates to those obtained by geostatistical inverse modeling where heterogeneity is assumed in all stages of estimation we use drawdown data from multiple pumping tests conducted at the test site in krauthausen germany the geometric mean values of transmissivity and storativity determined by type curve analysis are very close to those obtained by geostatistical inversion but the conventional approach failed to resolve the spatial variability of transmissivity in contrast the estimate from geostatistical inversion reveals more structure this indicates that the estimates of the type curve approaches can not be treated as pseudo local values concerning storativity both analysis methods show strong fluctuations because the variability of all terms making up the storativity is small we believe that the estimated variability of storativity is biased we examine the influence of measurement error on estimating structural parameters of covariance functions in the inversion we obtain larger correlation lengths and smaller prior variances if we trust the measured data less

analytical solutions to statistical moments for transient flow in two dimensional bounded randomly heterogeneous media 1 in this paper we derive analytical solutions to statistical moments for transient saturated flow in two dimensional bounded randomly heterogeneous porous media by perturbation expansions we first derive partial differential equations governing the zeroth order head h 0 and the first order head term h 1 where orders are in terms of the standard deviation of the log transmissivity we then solve h 0 and h 1 analytically both of which are expressed as infinite series the head perturbation h 1 is then used to derive expressions for autocovariance of the hydraulic head and the cross covariance between the log transmissivity and head the expressions for the mean flux and flux covariance tensor are formulated from the head moments based on darcy s law using numerical examples we demonstrate the convergence of these solutions we also examine the accuracy of these first order solutions by comparing them with solutions from both monte carlo simulations and the numerical moment equation method

existence of solutions bounded together with their first derivatives of some systems of nonlinear functional differential equations we give sufficient conditions for the system of nonlinear functional differential equations of the formx t 1 ax t f t x t x v 1 t x v k t x g 1 t x g l t where a is a matrix deta 0 v j t email protected j 1 t x f j 2 x f j m j t x t j 1 email protected and functions f r x r n k l 1 r n g j r r j 1 email protected f j i r xr n r j 1 email protected i 1 m email protected are continuous for the unique existence of continuously differentiable solutions x t which are bounded together with their first derivatives on r 0 and satisfy the condition lim t x t 1 ax t 0

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

on bounded continuously differentiable solutions with lipschitz first derivatives of a system of functional differential equations sufficient conditions for the unique existence of a continuously differentiable solution bounded together with its first derivative and whose first derivative satisfies the lipschitz condition of a boundary value problem of a system of functional differential equations with iterated deviating arguments dependent of unknown function are given

on solutions of a class of systems of nonlinear functional differential equations of neutral type with complicated deviations of an argument abstract sufficient conditions for the existence of a unique bounded solution together with its first derivative on r of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function and satisfying the condition lim t x t 1 x t 0 are given

mean travel time of conservative solutes in randomly heterogeneous unbounded domains under mean uniform flow 1 we derive a closed form expression for mean travel time of a conservative solute migrating under uniform in the mean flow conditions within an infinite stationary field with simple exponential correlation of the natural logarithm of hydraulic conductivity our expression is developed from a consistent second order expansion in y standard deviation of the log hydraulic conductivity of the equations for moments of travel time and trajectories of conservative solutes in two dimensional randomly nonuniform flows of guadagnini et al 2001 as such it is nominally valid for moderately heterogeneous fields with y2 1 its validity for larger heterogeneity degrees is tested against numerical monte carlo simulations our results clarify the nonlinear effect in the mean travel time with respect to distance that has been observed numerically and modeled empirically in the literature

correlation structure of flow variables for steady flow toward a well with application to highly anisotropic heterogeneous formations the study a continuation of that of indelman et al 1996 aims at deriving the second order moments of flow variables such as hydraulic head its gradient and the specific discharge for steady flow toward a fully penetrating well in a confined heterogeneous aquifer the log conductivity y ln k is modeled as a three dimensional stationary function of gaussian correlation of anisotropy ratio e by using first order approximations in 2y and e we derive the variance and the vertical integral scale of the piezometric head h of its radial gradient er and of the radial component of the specific discharge qr owing to the nonuniformity of the average flow these quantities are functions of the distance from the well it is shown that the variances of the head 2h and of its gradient 2er as well as the crossvariance e y between er and y vanish at the well whereas the discharge variance 2qr tends to the product between the log conductivity variance 2y and the squared mean discharge qr 2 this behavior pertains to a stratified formation surrounding the well far from the well 75 horizontal y integral scales i the head variance approaches a constant value for r 10i the moments 2er 2qr and ery tend to the corresponding values for uniform flow but with the local mean head gradient replacing the constant one the head vertical integral scale grows indefinitely with r whereas the vertical integral scale of the flux is larger by one log conductivity vertical scale than the one prevailing in uniform flow this latter property is explained by the presence of the source line which increases the correlations in the vertical direction the present results may be used in identifying the log conductivity statistical parameters from flowmeter velocity measurements in piezometers surrounding pumping or injecting wells

transmissivity upscaling in numerical aquifer models of steady well flow unconditional statistics 1 numerical solution of regional scale aquifer flow requires discretizing the transmissivity t typically the numerical element scale r and the transmissivity integral scale i are both of the order of hundreds of meters then the upscaled block transmissivity is random and its statistical moments depend on those of t on flow conditions and on r i modeling y ln t as a two dimensional normally distributed stationary random field we derive unconditional statistics of the upscaled ln accurate to the first order in y2 for strongly nonuniform flow in a circular block of radius r centered on a well of radius rw in an unbounded domain after a proper definition it was found that the ensemble mean is approximately but not exactly equal to ln tg the geometric mean and that the variance ratio 2 y2 which depends on rw i and the shape of the correlation drops slowly from unity for r i 0 to approximately 0 6 for r i 10 hence the variance of the upscaled transmissivity in radial flow is larger than that determined previously for uniform flow additionally defining the equivalent teq as a deterministic value for which the solution of the flow problem renders directly the mean upscaled head drop and specific discharge we find that in radial flow teq th the harmonic mean and grows slowly with increasing r i whereas for mean uniform flow teq tg application of the procedure is illustrated for an example of aquifer with selected values of parameters

existence of bounded solutions of a class of neutral systems of functional differential equations some results on the existence of bounded solutions together with their first derivatives of a class of neutral systems of functional differential equations with complicated deviations which extend and unify numerous results in the literature are proved

convexity properties of gradient maps abstract we consider the action of a real reductive group g on a kahler manifold z which is the restriction of a holomorphic action of a complex reductive group h we assume that the action of a maximal compact subgroup u of h is hamiltonian and that g is compatible with a cartan decomposition of h we have an associated gradient map p z p where g k p is the cartan decomposition of g for a g stable subset y of z we consider convexity properties of the intersection of p y with a closed weyl chamber in a maximal abelian subspace a of p our main result is a convexity theorem for real semi algebraic subsets y of z p v where v is a unitary representation of u

unique existence of periodic solutions of a class of systems of nonlinear difference equations with continuous argument some sufficient conditions which guarantee the unique existence of continuous periodic solutions of a class of systems of nonlinear difference equations with continuous argument are given here more important part in the proofs of presented results are descriptions of the methods for constructing these solutions which extend some methods in the literature

note on the existence of periodic solutions of a class of systems of differential difference equations abstract we prove a result on the existence of periodic solutions of a class of systems of differential difference equations partially solved with respect to the first order derivative by constructing it in a form of a series

stability of the zero solution of a family of functional differential equations abstract we give some sufficient conditions under which the zero solution of the next functional differential equation x t ax t bx 0 t j 1 k c j x j t f x t x k 1 t x k l t where a b c j j 1 k are real numbers f r l 1 r is a continuous function such that f 0 0 0 and j t c 2 0 j 0 k l is asymptotically stable

transient well type flows in heterogenous formations 1 the problem of averaging transient flows by sources of a given head boundary condition in heterogeneous formations of random conductivity is investigated the study generalizes the recently developed mathematical model of average transient nonuniform flow indelman 1996 tartakovsky and neuman 1998a the latter allows calculating the mean head for sources of flux boundary condition and as such is not applicable to modeling common well flows to account for the head condition the source term in the local flow model is modeled by a random function proportional to the hydraulic conductivity the local flow equations are further averaged in order to determine the mean flow variables it is shown that the effective conductivity cannot be defined for arbitrary initial head distributions this precludes deriving the mathematical model of an average transient flow with the head boundary condition however the average flow equations are derived for the important particular case of a uniformly distributed initial head the effective conductivity tensor is defined for an arbitrary heterogeneous formation and analyzed in detail for isotropic media it is shown that at the initial stage of the transient flow the effective conductivity is larger than the conductivity arithmetic mean the fundamental solution of the average flow equations mean green function corresponding to the mean head distribution due to the instantaneous injection of the unit mean amount of water through a point source is derived at first order in the conductivity variance and is compared with the mean head for the flux boundary condition

steady flow toward wells in heterogeneous formations mean head and equivalent conductivity we consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw figure 1 the hydraulic conductivity k is modeled as a three dimensional stationary random space function the two point covariance of y in k kg is of axisymmetric anisotropy with i and i the horizontal and vertical integral scales respectively and kg the geometric mean of k unlike previous studies which assumed constant flux the well boundary condition is of given constant head figure 1 the aim of the study is to derive the mean head h and the mean specific discharge q as functions of the radial coordinate r and of the parameters y2 e i i and rw i an approximate solution is obtained at first order in y2 by replacing the well by a line source of strength proportional to k and by assuming ergodicity i e equivalence between space averages over the vertical and h q ensemble means an equivalent conductivity keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge q as the actual one for the given head hw in the well and a given head in a piezometer at distance r from the well this definition corresponds to the transmissivity determined in a pumping test by an observer that measures hw andq the main result of the study is the relationship 19 keq ka 1 kefu where ka is the conductivity arithmetic mean and kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer the weight coefficient 10 has the simple approximate expression ln r i in r rw near the well 0 and keq ka which is easily understood since for rw i 1 the formation behaves locally like a stratified one in contrast far from the well 1 and keq kefu the flow being slowly varying there since ka kefu our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well however the difference between ka and kefu is small for highly anisotropic formations for which e 1 a nonlocal effective conductivity which depends only on the heterogeneous structure is derived in appendix a along the lines of indelman and abramovich 1994

solutions converging to zero of some systems of nonlinear functional differential equations with iterated deviating argument abstract some sufficient conditions for the existence of a p parameter family of solutions converging to zero of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function are given

existence of bounded solutions of some systems of nonlinear functional differential equations with complicated deviating argument abstract sufficient conditions for the existence of bounded solutions on the real line satisfying lipschitz s condition of a class of systems of nonlinear functional differential equations with complicated deviating argument dependent of unknown functions are given in this paper

type curve estimation of statistical heterogeneity 1 the analysis of pumping tests has traditionally relied on analytical solutions of groundwater flow equations in relatively simple domains consisting of one or at most a few units having uniform hydraulic properties recently attention has been shifting toward methods and solutions that would allow one to characterize subsurface heterogeneities in greater detail on one hand geostatistical inverse methods are being used to assess the spatial variability of parameters such as permeability and porosity on the basis of multiple cross hole pressure interference tests on the other hand analytical solutions are being developed to describe the mean and variance first and second statistical moments of flow to a well in a randomly heterogeneous medium we explore numerically the feasibility of using a simple graphical approach without numerical inversion to estimate the geometric mean integral scale and variance of local log transmissivity on the basis of quasi steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate by local log transmissivity we mean a function varying randomly over horizontal distances that are small in comparison with a characteristic spacing between pumping and observation wells during a test experimental evidence and hydrogeologic scaling theory suggest that such a function would tend to exhibit an integral scale well below the maximum well spacing this is in contrast to equivalent transmissivities derived from pumping tests by treating the aquifer as being locally uniform on the scale of each test which tend to exhibit regional scale spatial correlations we show that whereas the mean and integral scale of local log transmissivity can be estimated reasonably well based on theoretical ensemble mean variations of head and drawdown with radial distance from a pumping well estimating the log transmissivity variance is more difficult we obtain reasonable estimates of the latter based on theoretical variation of the standard deviation of circumferentially averaged drawdown about its mean

cartan decomposition of the moment map we investigate a class of actions of real lie groups on complex spaces using moment map techniques we establish the existence of a quotient and a version of luna s slice theorem as well as a version of the hilbert mumford criterion a global slice theorem is proved for proper actions we give new proofs of results of mostow on decompositions of groups and homogeneous spaces

semistable points with respect to real forms we consider actions of real lie subgroups g of complex reductive lie groups on kahlerian spaces our main result is the openness of the set of semistable points with respect to a momentum map and the action of g

equivalent transmissivity of heterogeneous leaky aquifers for steady state radial flow 1 the purpose of this study is to estimate the equivalent transmissivity for two dimensional steady state radial flow toward a fully penetrating well in a heterogeneous leaky aquifer the log transmissivity of the pumped aquifer is modeled as a multivariate random spatial function with stationary mean and exponential semivariogram while the aquitard conductance is assumed to be spatially uniform an approximate expression for the upscaled or block transmissivity defined as the transmissivity of an equivalent homogeneous leaky aquifer system with the same pumping is proposed the proposed relation expresses the upscaled transmissivity as a weighted average of the log transforms of the point transmissivity values monte carlo simulations are used to numerically validate the proposed upscaling expression for wide ranges of aquifer parameters the relation of the weighing function to the statistical parameters of the transmissivity field is discussed the first and second moments of the upscaled transmissivity are also estimated as a function of the problem parameters the expected value of the equivalent transmissivity is shown to increase from the geometric mean to the arithmetic mean as the aquitard conductance increases from zero to infinity

convexity properties of the moment mapping re examined abstract consider a hamiltonian action of a compact lie group on a compact symplectic manifold a theorem of kirwan s says that the image of the momentum mapping intersects the positive weyl chamber in a convex polytope i present a new proof of kirwan s theorem which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold it also applies to some interesting classes of noncompact or singular hamiltonian spaces such as cotangent bundles and complex affine varieties

on distinguished orbits of reductive representations abstract let g be a real reductive lie group and let g gl v be a real reductive representation of g with restricted moment map m g v 0 g in this work we introduce the notion of nice space of a real reductive representation to study the problem of how to determine if a g orbit is distinguished i e it contains a critical point of the norm squared of m g we give an elementary proof of the well known convexity theorem of atiyah guillemin sternberg in our particular case and we use it to give an easy to check sufficient condition for a g orbit of an element in a nice space to be distinguished in the case where g is algebraic and is a rational representation the above condition is also necessary making heavy use of recent results of michael jablonski obtaining a generalization of nikolayevsky s nice basis criterion we also provide useful characterizations of nice spaces in terms of the weights of finally some applications to ternary forms are presented

on q difference asymptotic solutions of a system of nonlinear functional differential equations the asymptotic behavior and the existence of a continuously differentiable solution bounded together with its first derivative of a system of nonlinear functional differential equations is studied here

three dimensional numerical analysis of steady state pumping tests in heterogeneous confined aquifers 1 we study three dimensional steady state flow field generated by a fully penetrating well extracting a constant water discharge from a confined aquifer of constant thickness the hydraulic conductivity k is modeled as an axisymmetric stationary random space function and the flow equation is solved by using the seven node finite volume numerical scheme in view of application in aquifer characterization we analyze how the equivalent conductivity keq varies as a function of the distance from the extraction well the numerical simulations showed that the boundary condition at the well exerts a large impact on keq and that the first order solution in y2 obtained by indelman et al 1996 is valid up to y2 0 5 for higher values of y2 the first order solution overestimates keq with differences that for a given y2 reduce progressively as the formation becomes more anisotropic a simple inverse procedure based on keq obtained through cross well interference is finally proposed and tested with the objective of estimating the parameters of the geostatistical model of variability of the hydraulic conductivity a few parametric examples showed that a suitable interpretation of pumping tests can be used to safely estimate parameters such as the geometric mean of the hydraulic conductivity its variance and although with larger uncertainty the horizontal integral scale while the inference of the anisotropy ratio is highly uncertain and error prone

exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow received 3 august 2009 revised 12 july 2010 accepted 21 july 2010 published 4 november 2010 1 we present novel equations for the exact sensitivity matrix of the ensemble mean hydraulic head under steady state groundwater flow conditions these equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads our formulation allows considerable improvement of the methodology proposed by hernandez et al 2003 2006 and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters the spatial distribution of the natural logarithm y of conductivity is parameterized within the pilot points framework whereas prior values of y at pilot points are obtained by a variant of kriging posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads the maximum likelihood function also includes a regularization term by means of a synthetic example and upon adopting formal model information criteria we explore the influence of 1 the number of pilot points and 2 the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying y variogram we find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the kashyap s bayesian measure

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

an investigation of the validity of first order stochastic dispersion theories in isotropie porous media monte carlo simulations are used to 1 investigate the accuracy of approximations that are implicit in first order stochastic dispersion theories and 2 identify the accuracy limits of first order dispersion theories in isotropic porous media the fickian theory of gelhar and axness 1983 as well as the fickian and non fickian theories of dagan 1984 and neuman and zhang 1990 are investigated all monte carlo simulations are in three dimensions confidence limits of ensemble averaged monte carlo results in isotropic porous media are established for 0 1 y 1 5 these results showed that first order theoretical estimates of the eulerian velocity covariance function are quite accurate for y 1 theoretical estimates of the non fickian longitudinal dispersivity do not deviate significantly from theory for at least y 1 5 theoretical estimation of the transverse dispersivity is limited to y 1 and the fickian longitudinal dispersivity is overestimated by the theory of gelhar and axness 1983 of all first order dispersion theories the theory of dagan 1984 is most robust in estimating the dispersivity tensor

spatial averaging of transmissivity in heterogeneous fields with flow toward a well steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical empirical approach the objective of the study is to determine the effective transmissivity of a square field with a constant head external boundary and a constant rate well at its center transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function block transmissivities are defined using an empirical scaling up of point support values within the field the scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field the principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries the geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale exact expressions are derived for the ensemble moments of block averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known the ensemble mean of block averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation this result contradicts earlier published work numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

on a nonlinear generalized max type difference equation abstract the boundedness character of positive solutions of the following max type difference equation x n max a x n 1 p x n k r n n 0 where k n 1 the parameters a and r are positive and p is a nonnegative real number is studied in this paper our main results considerably improve results appearing in the literature

beyond thiem a new method for interpreting large scale pumping tests in heterogeneous aquifers 1 the heterogeneous character of aquifers prevents a consistent interpretation of pumping tests based on thiem s formula which has been developed for homogeneous aquifers we present a new mathematical method for interpreting an ensemble and single large scale pumping tests in heterogeneous confined isotropic aquifers by combining a multiscale technique called coarse graining with renormalization group analysis we derive an explicit formula for the coarse grained aquifer transmissivity the coarse grained transmissivity is dependent on the radial distance from the well close to the well it equals the local transmissivity in single pumping tests and the harmonic mean of the heterogeneous transmissivity field in ensemble pumping tests in the far field of the well the coarse grained transmissivity approaches the geometric mean the transition between both regimes is mainly determined by the transmissivities correlation length this relation is used for an inversely estimate of the statistical properties of the transmissivity such as its geometric mean in addition this method allows for determining aquifer structures characterized by a correlation length from a single pumping test while the variance can be estimated from pumping tests with ensemble averaged hydraulic heads

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

erratum transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations water resources research 1998 34 1 1 12

ergodicity of pumping tests 1 standard interpretations of pumping tests in heterogeneous formations rely on effective representations of porous media which replace spatially varying hydraulic properties with their constant counterparts averaged over the support volume of a test rigorous approaches for deriving representative effective apparent upscaled etc parameters employ either ensemble or spatial averaging we derive a set of conditions under which these two paradigms yield identical results we refer to them as conditions for the ergodicity of pumping tests this allows one to use stochastic approaches to estimate the statistics of the spatial variability of hydraulic parameters on scales smaller than the support volume of a pumping test

a stochastic analysis of pumping tests in laterally nonuniform media conventional pumping test analysis methodology assumes that aquifer transmissivity is invariant in space the ramifications of this assumption are examined for hypothetical units whose variations in transmissivity are considered to be reasonable representations of variations possible in natural systems the dependence of pumping test transmissivity on spatial angular and radial and temporal location of observations and the method of drawdown analysis is assessed the dependence on the angular position of an observation well appears of little significance dependence on radial position however can be strong the dependence on the interval of time used in the pumping test analysis is only important in highly variable systems methods of drawdown analysis that yield identical estimates in uniform units yield differing estimates in nonuniform ones as a result of a difference in data fitting procedures a reasonable estimate of transmissivity at a regional scale can be obtained from a cooper jacob analysis applied at a large duration of pumpage in general conventional approaches for pumping test analysis should be viable in the nonuniform aquifers of the type considered here

asymptotic behavior of solutions of a system of q functional difference equations abstract the asymptotic behavior and the existence of a bounded lipschitz continuous solution of a system of nonlinear q functional difference equations on a half interval of the real line are studied here the main results compare the behavior of such a solution of the system with bounded solutions of a linear q difference equation

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

kahlerian potentials and convexity properties of the moment map since kostant proved his convexity theorems for torus actions on ag varieties ko there have been numerous contributions to this subject in the more general symplectic and k ahlerian setting for example let k be a compact lie group acting in a hamiltonian fashion on a connected compact symplectic manifold x then the intersection x of the image of the moment map x liek with a positive weyl chamber t is convex thus x k x where x is a natural convex section this was proved by atiyah and guillemin sternberg a g s in the abelian case i e where k t is a compact torus by mumford m for x projective algebraic with an integral k ahlerian structure and k not necessarily abelian and in its nal form in the compact symplectic case by kirwan k a precise description of x x t in the compact k ahlerian framework can be given in terms of the image of the components of the set of t xed points in x in the projective case where the k ahlerian structure comes from a projective embedding there are explicit connections to the representations theory of g kc which are due to brion and to mumford ness b n the purpose of the present paper is to prove convexity properties of in the setting of non compact k ahlerian spaces note that by removing an appropriate k invariant subset from x one can easily construct non convex images from convex ones on the other hand hilgert neeb plank proved h n p that convexity holds for a proper moment map x liek another convexity result has been obtained by sjamaar for x a ne and the restriction of the moment map of a representation s2 here the representation theoretical approach of brion is used now let x be an irreducible complex space endowed with a holomorphic action of a complex reductive group g assume that a maximal

on a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results in the literature