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

stochastic smoothed profile method for modeling random roughness in flow problems we present an efficient computational method to model fluid flow in the presence of random wall roughness a random flow domain is represented by a stochastic indicator function having a smoothed profile perpendicular to roughness and the random domain is discretized with a fixed non conformal grid this procedure introduces a stochastic force into the navier stokes equations and modifies the boundary conditions at the fluid solid interface we employ a high order semi implicit splitting scheme implemented in the context of a spectral hp element method in order to discretize the physical domain the stochastic roughness is treated as a second order autoregressive process that is represented by a karhunen loeve expansion a multi element probabilistic collocation method is employed to solve the resulting stochastic navier stokes equations this method is applied to simulate external flow past a rough cylinder and internal stokes flow between two parallel plates with random wall roughness in the first problem we develop an analytical solution for the asymptotic behavior of the lift coefficient cl to verify the results in the second test case we compare the mean and the standard deviation of the velocity field to those obtained from a different method called stochastic mapping approach sma developed by tartakovsky and xiu 2006

homogenous projective factors for actions of semi simple lie groups we analyze the structure of a continuous or borel action of a connected semi simple lie group g with finite center and real rank at least 2 on a compact metric or borel space x using the existence of a stationary measure as the basic tool the main result has the following corollary let p be a minimal parabolic subgroup of g and k a maximal compact subgroup let be a p invariant probability measure on x and assume the p action on x is mixing then either is invariant under g or there exists a proper parabolic subgroup q g and a measurable g equivariant factor map x g q m where k k dk and m is the k invariant measure on g q furthermore the extension has relatively g invariant measure namely x is induced from a mixing probability measure preserving action of q

state maps for linear systems modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential algebraic equations as such state space models are not a natural starting point for modeling while they have utmost importance in the simulation and control phase the paper addresses the problem of computing state variables for systems of linear differential algebraic equations of various forms the point of view from which the problem is considered is the behavioral one as put forward in j c willems automatica j ifac 22 1986 pp 561 580 dynamics reported 2 1989 pp 171 269 ieee trans automat control 36 1991 pp 259 294

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

brief transfer equivalence and realization of nonlinear higher order input output difference equations two fundamental modelling problems in nonlinear discrete time control systems are studied using the language of differential forms the discrete time nonlinear single input single output systems to be studied are described by input output i o difference equations i e a high order difference equation relating the input the output and a finite number of their time shifts a new definition of equivalence is introduced which generalizes the notion of transfer equivalence well known for the linear case our definition is based upon the notion of an irreducible differential form of the system and was inspired by the analogous definition for continuous time systems the second problem to be addressed is the realization problem the i o difference equation is assumed to be in the irreducible form so that one can obtain an accessible and observable realization necessary and sufficient conditions are given for the existence of a local state space realization of the irreducible i o difference equation these conditions are formulated in terms of the integrability of certain subspaces of one forms classified according to their relative degree the sufficiency part of the proof gives a constructive procedure up to finding the integrating factors and integration of the set of one forms for obtaining a locally observable and accessible state space system if the system is not in the irreducible form one has first to apply the reduction procedure to transform the system into the irreducible form

generic stationary measures and actions let g be a countably infinite group and let be a generating probability measure on g we study the space of stationary borel probability measures on a topological g space and in particular on z g where z is any perfect polish space we also study the space of stationary measurable g actions on a standard nonatomic probability space equip the space of stationary measures with the weak topology when has finite entropy we show that a generic measure is an essentially free extension of the poisson boundary of g when z is compact this implies that the simplex of stationary measures on z g is a poulsen simplex we show that this is also the case for the simplex of stationary measures on 0 1 g we furthermore show that if the action of g on its poisson boundary is essentially free then a generic measure is isomorphic to the poisson boundary next we consider the space of stationary actions equipped with a standard topology known as the weak topology here we show that when g has property t the ergodic actions are meager we also construct a group g without property t such that the ergodic actions are not dense for some finally for a weaker topology on the set of actions which we call the very weak topology we show that a dynamical property e g ergodicity is topologically generic if and only if it is generic in the space of measures there we also show a glasner king type 0 1 law stating that every dynamical property is either meager or residual

a structure theorem for actions of semisimple lie groups we consider a connected semisimple lie group g with finite center an admissible probability measure on g and an ergodic g u space x v we first note lemma 0 1 that x v has a unique maximal projective factor of the form g q vo where q is a parabolic subgroup of g and then prove 1 theorem 1 if every noncompact simple factor of g has real rank at least two then the maximal projective factor is nontrivial unless v is a g

modeling of nonlinear discrete time systems from input output data abstract an algorithm is presented that produces a polynomial state affine model for a discrete time nonlinear system the method uses only input output information and is based on a difference equation approximation to the input output map from the difference equation a behaviour matrix is constructed to be used in the state affine realization algorithm the difference equation approach requires the estimation of fewer parameters than the usual volterra series a state space model is then obtained directly from the difference equation as a byproduct a method for calculating the volterra series directly from the difference equation is obtained practical applications to nonlinear systems are included

mechanics of heterogeneous porous media with several spatial scales by extending the theory of homogenization we consider a heterogeneous porous medium whose material structure is characterized by multiple periodicity over several disparate length scales because for geological and engineering applications the driving force e g the pressure gradient and the global motion of primary interest is usually at a scale much larger than the largest of structural periodicity we derive the phenomenological equations of motion by using the perturbation technique of multiple scales all the effective coefficients are defined by boundary value problems on unit cells in smaller scales and no constitutive assumptions are added aside from the basic equations governing the mechanics of the pore fluid and the solid matrix in part i the porous matrix is assumed to be rigid the case of three scales is treated first symmetry and positiveness of the effective darcy permeability tensor are proven extensions to four and more scales are then discussed in part ii we allow the solid matrix to be deformable and deduce the equations of consolidation of a two phase medium the coupling of fluid flow and the quasi static elastic deformation of the matrix is considered when there are three disparate length scales effective coefficients of various kinds are deduced in terms of cell problems on the scale of the pores several general symmetry relations as well as specific properties of a medium composed of layers of porous matrix are discussed

perturbation theory for effective diffusivity in random gradient flows we investigate a result for the effective diffusivity of particles in a random gradient flow previously obtained by an intuitively plausible renormalization group argument and very accurately verified by numerical simulation we show that to two loop order the result is consistent with a direct perturbation theory calculation to the same order in perturbation theory we also derive a ward identity which guarantees the equality of the ratio of the effective diffusivity to the renormalized coupling with the ratio of the corresponding bare values the invariance of this ratio under renormalization was an important feature of the successful renormalization group calculation

minimal realisation of bilinear and quadratic input output difference equations in state space form this article studies the realisability property of discrete time bilinear and quadratic input output i o equations in the classical state space form constraints on the parameters of the i o model are suggested that lead to realisable models using new formulae for computing basis vectors of certain vector spaces of differential one forms we present in this article the complete list of the third and fourth order realisable i o bilinear models and a new realisable subclass of an arbitrary order is suggested moreover we provide the sufficient conditions of the second and third order realisable i o quadratic models respectively all the developed theory and algorithms are illustrated by means of several examples

linearization of discrete time systems the algebraic formalism developed in this paper unifies the study of the accessibility problem and various notions of feedback linearizability for discrete time nonlinear systems the accessibility problem for nonlinear discrete time systems is shown to be easy to tackle by means of standard linear algebraic tools whereas this is not the case for nonlinear continuous time systems in which case the most suitable approach is provided by differential geometry the feedback linearization problem for discrete time systems is recasted through the language of differential forms in the event that a system is not feedback linearizable the largest feedback linearizable subsystem is characterized within the same formalism using the notion of derived flag of a pfaffian system a discrete time system may be linearizable by dynamic state feedback though it is not linearizable by static state feedback necessary and sufficient conditions are given for the existence of a so called linearizing output which in turn is a sufficient condition for dynamic state feedback linearizability

classical state space realizability of input output bilinear models this paper studies the realizability property of bilinear input output i o models in the classical state space form constraints on the parameters of the bilinear i o model are suggested that lead to realizable models the complete list of 2nd and 3rd order realizable input output bilinear models together with the corresponding state equations is given in the general case some subclasses of realizable bilinear models together with their state space realizations are presented including the diagonal bilinear model and the special subclass of the superdiagonal bilinear model

on classical state space realisability of quadratic input output differential equations this article studies the realisability property of continuous time quadratic input output i o equations in the classical state space form constraints on the parameters of the quadratic i o model are suggested that lead to realisable models the complete list of second and third order realisable i o quadratic models is given and two subclasses of the n th order realisable i o quadratic systems are suggested our conditions rely basically upon the property that certain combinations of coefficients of the i o equations are zero or not zero we provide explicit state equations for realisable second order quadratic i o equations and for one realisable subclass of quadratic i o equations of arbitrary order

a perturbation expansion for diffusion in a random velocity field we present a formal perturbation expansion for the large scale effective drift velocity and diffusion matrix of a medium with stationary random velocity v x and constant nonrandom diffusion matrix a on a small scale if mu denotes the mean of v x and we express v x as mu varepsilon u x then the expansion is in powers of varepsilon with mu u and a fixed we obtain explicit expressions up to second order in varepsilon which generalize the standard formulae to the case a ne 0

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

stabilizers for ergodic actions of higher rank semisimple groups let g be a connected semisimple lie group with finite center and r rank 2 suppose that each simple factor of g either has r rank 2 or is locally isomorphic to sp l n or f4 20 we prove that any faithful irreducible properly ergodic finite measure preserving action of g is essentially free we extend the result to reducible actions and actions of lattices

some realizations of algorithms for nonlinear input output systems this paper develops an algorithm which given an input output differential equation produces under appropriate circumstances a parametrization of the observation space of the underlying state space system the paper extends previous results by the authors to a more general setting so that state space systems in which the controls enter nonlinearly can be accommodated and most importantly the paper deals with implicitly defined systems at both the input output and state space level fliess and hasler have demonstrated the importance of implicit input output models which yield only local state space descriptions various realizations are obtained depending on the properties of the derived candidate observation space examples of the algorithm are described with the aid of an implementation using the computer algebra system axiom

the use of field theoretic methods for the study of flow in a heterogeneous porous medium the effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined it is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system it is shown that the problem may be written as a field theory which allows a perturbation series to be expressed by diagrammatic means this allows the calculation of effective permeability the mean pressure and the pressure variance the method as well as recovering familiar results gives a formal means of improving the approximation and approaching more complex systems

numerical solutions of moment equations for flow in heterogeneous composite aquifers 1 we analyze flow in heterogeneous media composed of multiple materials whose hydraulic properties and geometries are uncertain our analysis relies on the composite media theory of winter and tartakovksy 2000 2002 which allows one to derive and solve moment equations even when the medium is highly heterogeneous we use numerical solutions of darcy flows through a representative composite medium to investigate the robustness of perturbation approximations in porous medium with total log conductivity variances as high as 20 we also investigate the relative importance of the two sources of uncertainty in composite media material properties and geometry in our examples the uncertain geometry by itself captures the main features of the mean head estimated by the full composite model even when the within material conductivities are deterministic however neglecting randomness within materials leads to head variance estimates that are qualitatively and quantitatively wrong we compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields with deterministic trends we demonstrate that models with a deterministic trend can be expected to give a poor estimate of the statistics of head

random walks on random coset spaces with applications to furstenberg entropy we determine the range of furstenberg entropy for stationary ergodic actions of nonabelian free groups by an explicit construction involving random walks on random coset spaces

unimodularity of invariant random subgroups an invariant random subgroup h g is a random closed subgroup whose law is invariant to conjugation by all elements of g when g is locally compact and second countable we show that for every invariant random subgroup h g there almost surely exists an invariant measure on g h equivalently the modular function of h is almost surely equal to the modular function of g restricted to h we use this result to construct invariant measures on orbit equivalence relations of measure preserving actions additionally we prove a mass transport principle for discrete or compact invariant random subgroups

state space realization of bilinear continuous time input output equations this paper studies the realizability property of continuous time bilinear input output i o equations in the classical state space form constraints on the parameters of the bilinear i o model are suggested that lead to realizable models the paper proves that the 2nd order bilinear i o differential equation unlike the discrete time case is always realizable in the classical state space form the complete list of 3rd and 4th order realizable i o bilinear models is given and two subclasses of realizable i o bilinear systems are suggested our conditions rely basically upon the property that certain combinations of coefficients of the i o equations are zero or not zero we provide explicit state equations for all realizable 2nd and 3rd order bilinear i o equations and for one realizable subclass of bilinear i o equations of arbitrary order

analysis of flow through heterogeneous random aquifers by the method of embedding matrix 1 steady flow the statistical structures of the hydraulic conductivity head gradient specific discharge and head fields are analyzed with the aid of a simplified model consisting of a collection of spherical in three dimensions and cylindrical in two dimensions blocks of different conductivity and diameter set at random and independently in space results of a previous work for the effective conductivity and unbounded formations dagan 1979 are recovered and given a rational derivation the influence of the aquifer boundary and nonuniform average flows are examined for the first time the growth of the head variance with distance from boundary in two dimensional flows is compared with that of the monte carlo simulations of smith and freeze 1979

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

dynamics of free surfaces in random porous media we consider free surface flow in random porous media by treating hydraulic conductivity of a medium as a random field with known statistics we start by recasting the boundary value problem in the form of an integral equation where the parameters and domain of integration are random our analysis of this equation consists of expanding the random integrals in taylor s series about the mean position of the free boundary and taking the ensemble mean to quantify the uncertainty associated with such predictions we also develop a set of integro differential equations satisfied by the corresponding second ensemble moments the resulting moment equations require closure approximations to be workable we derive such closures by means of perturbation expansions in powers of the variance of the logarithm of hydraulic conductivity though this formally limits our solutions to mildly heterogeneous porous media our analytical solutions for one dimensional flows demonstrate that such perturbation expansions may remai

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

on bilinear estimation and control discrete single input single output bilinear systems with constant parameters are considered first the minimum variance regulator for known parameters is determined then the least squares parameter estimation method is applied and is seen to have certain desirable properties finally the least squares parameter estimator is combined with the minimum variance regulator for the estimated model to produce a bilinear self tuning regulator blstr it is found that though the estimation may be biased under certain assumptions the blstr will converge to the minimum variance control law

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

comparison of lpv and nonlinear system theory a realization problem abstract the paper explores the utilization of the nonlinear realization theory to address the problem of transforming linear parameter varying input output lpv io equations into a state space form with static dependence on the so called scheduling parameter the necessary and sufficient solvability conditions are given and three additional subclasses of lpv io equations are suggested that are guaranteed to have a realization of the considered type

kesten s theorem for invariant random subgroups an invariant random subgroup of the countable group is a ran dom subgroup of whose distribution is invariant under conjuga tion by all elements of we prove that for a nonamenable invariant random subgroup h the spectral radius of every finitely supported random walk on i s strictly less than the spectral radius of the corresponding random walk on h this generalizes a result of kesten who proved this for normal subgroups as a byproduct we show that for a cayley graph g of a linear group with no amenable normal subgroups any sequence of finite quotients of g that spectrally approximates g converges to g in benjamini schramm convergence in particular this implies that infinite sequences of finite d regular ramanujan schreier graphs have essentially large girth

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

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

stationary measures and equidistribution for orbits of nonabelian semigroups on the torus let be a semigroup of d d nonsingular integer matrices and consider the action of on the torus t we assume throughout that the action is strongly irreducible there is no subtorus invariant under a finite index subsemigroup of the strong irreducibility assumption in particular implies that acts ergodically on t equipped with the lebesgue measure m therefore the orbit of lebesgue almost every x t is dense and in an appropriate sense even becomes equidistributed however when is cyclic there is a set of full hausdorff dimension of exceptional points x for which x fails to be dense when is bigger the distribution of individual orbits can be expected to be much more restrictive an important result in this direction is due to furstenberg who showed for d 1 in which case 1 an abelian semigroup never satisfies condition 1 indeed the group generated by a semigroup satisfying 1 is nonamenable assumption 2 is a technical condition which is in particular satisfied when is a zariski dense semigroup of sld z 16