morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

removal of boundary singularities of pseudo holomorphic curves with lagrangian boundary conditions we give a proof of the theorem of removing isolated singularities of pseudo holomorphic curves with lagrangian boundary conditions and bounded symplectic area the proof is a combination of some lp type estimates standard techniques of geometric p d e and some ideas from symplectic geometry and calibration theory

upper continuous and compactly generated properties of lattice of l fuzzy modules in this note it is shown that the lattice of l fuzzy modules of a given module is upper continuous and under a suitable condition it will be compactly generated

on the self averaging of dispersion for transport in quasi periodic random media in this study we present a numerical analysis for the self averaging of the longitudinal dispersion coefficient for transport in heterogeneous media this is done by investigating the mean square sample to sample fluctuations of the dispersion for finite times and finite numbers of modes for a random field using analytical arguments as well as numerical simulations we consider transport of point like injections in a quasi periodic random field with a gaussian correlation function in particular we focus on the asymptotic and pre asymptotic behaviour of the fluctuations with the aid of a probability density function for the dispersion and we verify the logarithmic growth of the sample to sample fluctuations as earlier reported in eberhard 2004 j phys a math gen 37 2549 71 we also comment on the choice of the relevant parameters to generate quasi periodic realizations with respect to the self averaging of transport in statistically homogeneous gaussian velocity fields

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

lifting morita equivalence to categories of fuzzy modules abstract in this paper we show that two rings r and s have equivalent categories of fuzzy left modules r fzmod and s fzmod if and only if r and s are morita equivalent furthermore any equivalence between r fzmod and s fzmod lifts up to natural isomorphism an equivalence between r mod and s mod in the process we introduce and study the basic properties of tensor products of fuzzy modules and tensor products of fuzzy maps

lagrangian floer theory on compact toric manifolds i we introduced the notion of weakly unobstructed lagrangian submanifolds and constructed their potential function po purely in terms of a model data in fooo3 in this article we carry out explicit calculations involving po on toric manifolds and study the relationship between this class of lagrangian submanifolds with the earlier work of givental g1 which advocates that the quantum cohomology ring is isomorphic to the jacobian ring of a certain function called the landau ginzburg superpotential combining this study with the results from fooo3 we also apply the study to various examples to illustrate its implications to symplectic topology of lagrangian fibers of toric manifolds in particular we relate it to the hamiltonian displacement property of lagrangian fibers and to entov polterovich s symplectic quasi states

multiplicity of lagrangian orbits on symmetric star shaped hypersurfaces abstract in this paper the multiplicity of lagrangian orbits on c 2 smooth compact symmetric star shaped hypersurfaces with respect to the origin in r 2 n is studied we prove an infinitely many existence result via z 2 index theory this is a multiplicity result about the arnold chord in some sense

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

hamiltonian pseudo rotations of projective spaces the main theme of the paper is the dynamics of hamiltonian diffeomorphisms of mathbb c mathbb p n with the minimal possible number of periodic points equal to n 1 by arnold s conjecture called here hamiltonian pseudo rotations we prove several results on the dynamics of pseudo rotations going beyond periodic orbits using floer theoretical methods one of these results is the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points partially extending a theorem of le calvez and yoccoz and of franks and misiurewicz to higher dimensions the other is a strong variant of the lagrangian poincare recurrence conjecture for pseudo rotations we also prove the c 0 rigidity of pseudo rotations with exponentially liouville mean index vector this is a higher dimensional counterpart of a theorem of bramham establishing such rigidity for pseudo rotations of the disk

numerical investigations on ergodicity of solute transport in heterogeneous aquifers 1 darcy velocities for lognormal hydraulic conductivity with small variance and finite correlation length were approximated by periodic random fields accurate simulations of two dimensional advection dispersion processes were achieved with the global random walk algorithm using 1010 particles in every transport realization reliable statistical estimations were obtained by averaging over 256 realizations the main result is a numerical evidence for the mean square convergence of the actual concentrations to the macrodispersion process predicted by a known limit theorem for small initial plumes the ergodic behavior can be expected after thousands of advection timescales when the deviation from the theoretical prediction of the cross section space averaged concentration monotonously decays and falls under 20 the increase of the transverse dimension of the initial plume slows down the approach to the quasi ergodic state and has a nonlinear effect on the variability of the actual concentrations and dispersivities

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

stratified disordered media exact solutions for transport parameters and their self averaging properties we investigate the transport of a passive tracer in a two dimensional stratified random medium with flow parallel and perpendicular to the strata assuming a gaussian random flow with a gaussian correlation function it is not only possible to derive exact expressions for the temporal behaviour of the dispersion coefficients characterizing the tracer transport but also to investigate the self averaging properties of these quantities we give explicit results for the dispersion coefficient and its mean square fluctuations as a function of time as it turns out the sample to sample fluctuations which are encoded in the latter quantity remain finite even in the asymptotic limit of infinite times which implies that the given two dimensional model is not self averaging

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

stochastic sensitivity a computable lagrangian uncertainty measure for unsteady flows uncertainties in velocity data are often ignored when computing lagrangian particle trajectories of fluids modeling these as noise in the velocity field leads to a random deviation from each traje

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

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

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

stochastic analysis of nonstationary subsurface solute transport 1 unconditional moments this paper applies stochastic methods to the analysis and prediction of solute transport in heterogeneous saturated porous media partial differential equations for three unconditional ensemble moments the concentration mean concentration covariance and velocity concentration cross covariance are derived by applying perturbation techniques to the governing transport equation for a conservative solute concentration uncertainty is assumed to be the result of unmodeled small scale fluctuations in a steady state velocity field the moment expressions which describe how each moment evolves over time and space resemble the classic deterministic advection dispersion equation and can be solved using similar methods a solution procedure based on a galerkin finite element algorithm is illustrated with a hypothetical two dimensional example for this example the required steady state velocity statistics are obtained from an infinite domain spectral solution of the stochastic groundwater flow equation the perturbation solution is shown to reproduce the statistics obtained from a monte carlo simulation quite well for a natural log conductivity standard deviation of 0 5 and moderately well for a natural log conductivity standard deviation of 1 0 the computational effort required for a perturbation solution is significantly less than that required for a monte carlo solution of acceptable accuracy sensitivity analyses conducted with the perturbation approach provide qualitative confirmation of a number of results obtained by other investigators for more restrictive special cases

on categories of fuzzy modules we show that for any ring r the category r fzmod of fuzzy left r modules is a top category and an additive category r fzmod has products coproducts kernels and cokernels but it is not an abelian category projective injective and free fuzzy left r modules are characterized

cup length estimate for lagrangian intersections abstract in this paper we consider the arnold conjecture on the lagrangian intersections of some closed lagrangian submanifold of a closed symplectic manifold with its image of a hamiltonian diffeomorphism we prove that if the hofer s symplectic energy of the hamiltonian diffeomorphism is less than a topology number defined by the lagrangian submanifold then the arnold conjecture is true in the degenerated nontransversal case

euler maclaurin with remainder for a simple integral polytope we give an euler maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope we prove this formula by elementary methods

on the topological properties of symplectic maps in this paper we show that symplectic maps have surprising topological properties in particular we construct an interesting metric for the symplectic diffeomorphism groups which is related but not obviously to the topological properties of symplectic maps and phase space geometry we also prove a certain number of generalised symplectic fixed point theorems and give an application to a hamiltonian system

fuzzy chain complex and fuzzy homotopy abstract in this study of fuzzy co chain complexes we define the notions of fuzzy co homology fuzzy exact sequence of fuzzy complexes and fuzzy homotopy then we construct the category of fuzzy complexes and then define an additive functor from this category and to the category of fuzzy modules finally we give some results about the fuzzy homotopy

persistent memory of diffusing particles the variance of the advection diffusion processes with variable coefficients is exactly decomposed as a sum of dispersion terms and memory terms consisting of correlations between velocity and initial positions for random initial conditions the memory terms quantify the departure of the preasymptotic variance from the time linear diffusive behavior for deterministic initial conditions the memory terms account for the memory of the initial positions of the diffusing particles numerical simulations based on a global random walk algorithm show that the influence of the initial distribution of the cloud of particles is felt over hundreds of dimensionless times in case of diffusion in random velocity fields with finite correlation range the particles forget the initial positions in the long time limit and the variance is self averaging with clear tendency toward normal diffusion

memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media 1 for transport in statistically homogeneous random velocity fields with properties that are routinely assumed in stochastic groundwater models the one particle dispersion i e second central moment of the ensemble average concentration for a point source is a memory free quantity independent of initial conditions nonergodic behavior of large initial plumes as manifest in deviations of actual solute dispersion from one particle dispersion is associated with a memory term consisting of correlations between initial positions and displacements of solute molecules reliable numerical experiments show that increasing the source dimensions has two opposite effects it reduces the uncertainty related to the randomness of center of mass but at the same time it yields large memory terms the memory effects increase with the source dimension and depend on its shape and orientation large narrow sources oriented transverse to the mean flow direction yield ergodic behavior with respect to the one particle dispersion of the longitudinal dispersion and nonergodic behavior of the transverse dispersion whereas for large longitudinal sources the longitudinal dispersion behaves nonergodically and the transverse dispersion behaves ergodically such memory effects are significantly large over hundreds of heterogeneity scales and should therefore be considered in practical applications for instance calibration of model parameters forecasting and identification of the contaminant source

coisotropic displacement and small subsets of a symplectic manifold we prove a coisotropic intersection result and deduce the following a lower bounds on the displacement energy of a subset of a symplectic manifold in particular a sharp stable energy gromov width inequality b a stable non squeezing result for neighborhoods of products of unit spheres c existence of a badly squeezable set in mathbb r 2n of hausdorff dimension at most d for every n 2 and d n d existence of a stably exotic symplectic form on mathbb r 2n for every n 2 e non triviality of a new capacity which is based on the minimal action of a regular coisotropic submanifold of dimension d

fuzzy finitely generated modules abstract we introduce the category of fuzzy modules and discuss the construction of fuzzy finitely generated modules characterizations of a sort of fuzzy modules in which the fuzzy value distribution is finite are partly obtained

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

the geometry of symplectic energy non squeezing theorem which says that it is impossible to embed a large ball symplectically into a thin cylinder of the form r2 x b2 where b2 is a 2 disc this led to hofer s discovery of symplectic capacities which give a way of measuring the size of subsets in symplectic manifolds recently hofer found a way to measure the size or energy of symplectic diffeomorphisms by looking at the total variation of their generating hamiltonians this gives rise to a bi invariant pseudo norm on the group ham m of compactly supported hamiltonian symplectomorphisms of the manifold m the deep fact is that this pseudo norm is a norm in other words the only symplectomorphism on m with zero energy is the identity map up to now this had been proved only for sufficiently nice symplectic manifolds and by rather complicated analytic arguments in this paper we consider a more geometric version of this energy which was first considered by eliashberg and hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary we prove by a simple geometric argument that both versions of energy give rise to genuine norms on all symplectic manifolds roughly speaking we show that if there were a symplectomorphism of m which had too little energy one could embed a large ball into a thin cylinder m x

some quantitative results in mathcal c 0 c 0 symplectic geometry this paper proceeds with the study of the mathcal c 0 symplectic geometry of smooth submanifolds as initiated in humiliere et al duke math j 164 4 767 799 2015 and opshtein ann sci ec norm super 42 5 857 864 2009 with the main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities our main result is that a symplectic homeomorphism may preserve and squeeze codimension 4 symplectic submanifolds mathcal c 0 flexibility while this is impossible for codimension 2 symplectic submanifolds mathcal c 0 rigidity we also discuss mathcal c 0 invariants of coistropic and lagrangian submanifolds proving some rigidity results and formulating some conjectures we finally formulate an eliashberg gromov mathcal c 0 rigidity type question for submanifolds which we solve in many cases our main technical tool is a quantitative h principle result in symplectic geometry

metric and isoperimetric problems in symplectic geometry introduction 411 1 some basic results in symplectic topology 413 2 capacity and symplectic reduction 414 3 volume estimates for lagrange submanifolds 416 3 1 the case of r 417 3 2 deformations of the zero section in cotangent bundles 419 3 3 generalization to the case of cp 422 4 an application to billiards 423 5 geometry of convex sets and periodic orbits 425 6 compensated compactness and closure of the symplectic group 426 appendix a a summary of symplectic geometry through generating functions 427 appendix b john s ellipsoid 428 references 430

prime fuzzy ideals over noncommutative rings in this paper we introduce prime fuzzy ideals over a noncommutative ring this notion of primeness is equivalent to level cuts being crisp prime ideals it also generalizes the one provided by kumbhojkar and bapat in 1993 which lacks this equivalence in a noncommutative setting semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes this allows us to introduce the fuzzy prime radical and contribute to establish the basis of a fuzzy noncommutative ring theory

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

fuzzy ideals of artinian rings abstract the object of this paper is to study fuzzy ideals over artinian rings among others we prove that a ring with unity is left right artinian if and only if every fuzzy left right ideal of r is finite valued this gives a characterization of artinian rings in terms of fuzzy ideals mukherjee and sen 1987 proved that if every fuzzy ideal of a ring with unity is finite valued then the ring is noetherian we give an example to show that the converse of this theorem is not true we also show that certain results of pan 1987 are not true

stochastic theory of field scale fickian dispersion in anisotropic porous media a three dimensional theory is described for field scale fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities the study relies partly on earlier work by the authors the attributes of which are briefly reviewed it leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media we express the dispersion tensor d as the sum of a local component d and a field scale component the local component is assumed to be independent of velocity which is most appropriate if it represents molecular diffusion and its principal terms are taken to act parallel and normal to the mean velocity vector the field scale component is written as where is a dispersivity tensor and we show that at large peclet numbers p the dispersivity tensor reduces to a single principal component parallel to the mean velocity regardless of how is oriented this result valid for arbitrary covariance functions of log hydraulic conductivity differs from that of l w gelhar and c l axness 1983 according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value they calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation or away from the stratification in typical layered media we show that this principal dispersivity is offset in the opposite direction at small and intermediate peclet numbers but rotates toward the mean velocity as p increases the largest eigen value is constant and dominated by field scale velocity fluctuations at large p values the other two eigen values diminish asymptotically in proportion to p 1 and are controlled by d as well as by field scale differential convection the range of small peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low permeability rocks or soils we show that at low p values all three principal dispersivities are proportional to p and thus is proportional to 2 a phenomenon reminiscent of taylor diffusion when the mean velocity is inclined to the axes of anisotropy the eigen values of are neither parallel nor normal to however since d is dominated by d at small peclet numbers the principal dispersion coefficients are asymptotically as p 0 parallel and normal to the mean velocity just like when p is large their maximum deviation from these directions occurs at intermediate p values

new topological field theories in two dimensions it is shown that two 1 1 dimensional 2d free abelian and self interacting non abelian gauge theories without any interaction with matter fields belong to a new class of topological field theories tfts these new theories capture together some of the key features of witten and schwarz types of tft because they are endowed with symmetries that are reminiscent of the schwarz type theories but their lagrangian density has the appearance of the witten type theories the topological invariants for these theories are computed on a 2d compact manifold and their recursion relations are obtained these new theories are shown to provide a class of tractable field theoretical models for the hodge theory in two dimensions of flat minkowski spacetime where there are no propagating degrees of freedom associated with the 2d gauge boson

lagrangian non squeezing and a geometric inequality we prove that if the unit codisc bundle of a closed riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest non trivial closed geodesic is at most half the area of the unit disc

multiplicity of characteristics with lagrangian boundary values on symmetric star shaped hypersurfaces abstract in this paper the multiplicity of lagrangian orbits on c 2 smooth compact symmetric star shaped hypersurfaces with respect to the origin in r 2 n is studied these lagrangian orbits begin from one lagrangian subspace and end on another an infinitely many existence result is proved via z 2 index theory this is a multiplicity result about the arnold chord conjecture in some sense and is a generalization of the problem about the multiplicity of lagrangian orbits beginning from and ending on the same lagrangian subspace which was considered in the authors previous paper f guo c liu multiplicity of lagrangian orbits on symmetric star shaped hypersurfaces nonlinear anal 69 4 2008 1425 1436

dispersion variance for transport in heterogeneous porous media 1 we study dispersion in heterogeneous porous media for solutes evolving from point like and extended source distributions in d 2 and d 3 spatial dimensions the impact of heterogeneity on the dispersion behavior is captured by a stochastic modeling approach that represents the spatially fluctuating flow velocity as a spatial random field we focus here on the sample to sample fluctuations of the dispersion coefficients about their ensemble mean for finite source sizes the definition of dispersion coefficients in single realizations is not unique we consider dispersion measures that describe the extension of the solute distribution as well as dispersion coefficients that quantify the solute spreading relative to injection points of the partial plumes that constitute the solute distribution while the ensemble averages of these dispersion quantities may be identical their fluctuation behavior is found to be different using a perturbation approach in the fluctuations of the random flow field we derive explicit expressions for the temporal evolution of the variances of the dispersion coefficients between realizations their evolution is governed by the typical dispersion time over the characteristic heterogeneity scale and the dimensions of the source distribution we find that the dispersion variance decreases toward zero with time in d 3 spatial dimensions while in d 2 it converges toward a finite long time value that is independent of the source dimensions

approximations for transport parameters and self averaging properties for point like injections in heterogeneous media we focus on transport parameters in heterogeneous media with a flow modelled by an ensemble of periodic and gaussian random fields the parameters are determined by ensemble averages we study to what extent these averages represent the behaviour in a single realization we calculate the centre of mass velocity and the dispersion coefficient using approximations based on a perturbative expansion for the transport equation and on the iterative solution of the langevin equation compared with simulations the perturbation theory reproduces the numerical results only poorly whereas the iterative solution yields good results using these approximations we investigate the self averaging properties the ensemble average of the velocity characterizes the behaviour of a realization for large times in both ensembles the dispersion coefficient is not self averaging in the ensemble of periodic fields for the gaussian ensemble the asymptotic dispersion coefficient is self averaging for finite times however the fluctuations are so large that the average does not represent the behaviour in a single realization

rough ideals under relations associated to fuzzy ideals in this paper we deal with the theory of rough ideals started in davvaz 2004 we show that the approximation spaces built from an equivalence relation compatible with the ring structure i e associated to a two sided ideal are too naive in order to develop practical applications we propose the use of certain crisp equivalence relations obtained from fuzzy ideals these relations make available more flexible approximation spaces since they are enriched with a wider class of rough ideals furthermore these are fully compatible with the notion of primeness semiprimeness the theory is illustrated by several examples of interest in engineering and mathematics