fractional brownian motion and fractional gaussian noise in subsurface hydrology a review presentation of fundamental properties and extensions recent studies have shown that fractional brownian motion fbm and fractional gaussian noise fgn are useful in characterizing subsurface heterogeneities in addition to geophysical time series although these studies have led to a fairly good understanding of some aspects of fbm fgn a comprehensive introduction to these stochastic fractal functions is still lacking in the subsurface hydrology literature in this paper efforts have been made to define fbm fgn and present a development of their mathematical properties in a direct yet rigorous manner use of the spectral representation theorem allows one to derive spectral representations for fbm fgn even though these functions do not have classical fourier transforms the discrete and truncated forms of these representations have served as a basis for synthetic generation of fbm fgn the discrete spectral representations are developed and various implications discussed in particular it is shown that a discrete form of the fbm spectral representation is equivalent to the well known weierstrass mandelbrot random fractal function although the full implications are beyond the scope of the present paper it is observed that discrete spectral representations of fbm constitute stationary processes even though fbm is nonstationary a new and general spectral density function is introduced for construction of complicated anisotropic 3 d fractals including those characterized by vertical fgn and horizontal fbm such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes finally some basic properties of fractional levy motion and concepts of universal multifractals which can be considered as generalizations of fbm fgn are reviewed briefly

multifractal anisotropic scaling of the hydraulic conductivity 1 we analyzed the scaling properties of the hydraulic conductivity k at three sites in northern america made borden and cape cod we found that k at all sites exhibits multifractality fractal and multiscaling in both the vertical and horizontal directions though the multiscaling was within a range smaller than that of the maximum distance between measurements in the vertical direction the k data for made borden and cape cod were multiscaling from 0 15 to 1 35 m 0 05 to 0 5 m and 0 15 to 0 9 m respectively they were multiscaling in the horizontal direction from 9 to 45 m 1 to 10 m and 1 to 17 m respectively the multiscaling was also anisotropic evidence of scaling was poorest for the horizontal direction of the made site and it spanned half an order of magnitude such results compel one to treat multifractality in the horizontal direction of made as supported by heuristic arguments rather than by pure statistical evaluation we fitted a multifractal model to the data and estimated its parameters we found the underlying statistics of all data to be non gaussian and the model capable of reproducing the probability distribution of k data especially the negative skewness of log k we also generated two dimensional isotropic multifractal fields illustrating the role of the parameters of the selected multifractal model

on the convergence in distribution of measurable multifunctions random sets normal integrands stochastic processes and stochastic infima the concept of the distribution function of a closed valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi convergence in distribution of normal integrands and stochastic processes in particular various compactness criteria are exhibited the connections with the classical convergence theory for stochastic processes are analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of donsker s theorem brownian motion as a limit of normalized random walks we also suggest the potential application of the theory to the study of the convergence of stochastic infima

on global leader following consensus of identical linear dynamic systems subject to actuator saturation abstract this paper studies the leader following consensus problem for a group of agents with identical linear systems subject to control input saturation we focus on two classes of linear systems neutrally stable systems and double integrator systems for neurally stable systems we establish that global consensus can be achieved by linear local feedback laws over a fixed communication topology and with proper choices of relative potential functions global consensus can also be achieved over a switching communication topology for double integrator systems we establish that global consensus can be achieved by linear local feedback laws over a fixed communication topology and with the help of a simple saturation function in the local feedback laws global consensus can also be achieved over a switching communication topology simulation results illustrate the theoretical results

differentiability of relations and differential stability of perturbed optimization problems several new concepts of differentiability of multifunctions are introduced special attention is paid to relations given by perturbed inequalities applications are given to the study of the value or marginal function of a perturbed nonlinear program

multifractal scaling of the intrinsic permeability existing fractal studies dealing with subsurface heterogeneity treat the logarithm of the permeability k as the variable of concern we treat k as a multifractal and investigate its scaling and fractality using measured horizontal k data from two locations in the united states the first data set was from a shoreline sandstone near coalinga california and the second was from an eolian sandstone goggin 1988 by applying spectral analyses and computing the scaling of moments of various orders using the double trace moment method lavallee 1991 lavallee et al 1992 we found that k is multiscaling i e scaling and multifractal we also found that the so called universal multifractal um schertzer and lovejoy 1987 model essentially a log levy multifractal was able to reproduce the multiscaling behavior reasonably well the um model has three parameters a s and h representing the multifractality index the codimension of the mean field and the distance to stationary multifractal respectively we found a 5 1 7 s 5 0 23 h 5 0 22 and a 5 1 6 s 5 0 11 h 5 0 075 for the shoreline and eolian data sets respectively the fact that a values were less than 2 indicates that the underlying statistics are non gaussian we generated stationary and nonstationary multifractals and illustrated the role of the um parameters on simulated fields studies that treated log k as the variable of concern have pointed out the necessity for large data records especially when the underlying distribution is levy stable our investigation revealed that even larger data records are required when treating k as a multifractal because log k is less intermittent or irregular than k

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

flexible scaling model for use in random field simulation of hydraulic conductivity a new fractal scaling model is proposed for use in aquifer heterogeneity modeling and simulation the new model which is obtained by subordination of fractional brownian motion provides a unifying framework for scaling models of heterogeneity in that it includes previous scaling models as special cases and can also be tuned continuously between these models choosing the subordinator to be a lognormal distribution results in a non gaussian scaling model with a moderate degree of variability that matches data better than previous fractal scaling models thus providing more accurate heterogeneity representations for use in stochastic flow and transport predictions two new stochastic simulation algorithms are constructed from the new models one based on sequential simulation algorithms and the other based on probability field simulation both algorithms can be made conditional on available measurements

a fractal based stochastic interpolation scheme in subsurface hydrology real porosity and hydraulic conductivity data do not vary smoothly over space so an interpolation scheme that preserves irregularity is desirable such a scheme based on the properties of fractional brownian motion fbm and fractional gaussian noise fgn is presented following the methodology of hewett 1986 the authors test for the presence of fgn in a set of 459 hydraulic conductivity k measurements the use of rescaled range analysis strongly indicated the presence of fgn when applied to the natural logs of the k data and the resulting hurst coefficient h was determined to be 0 82 this h value was then used along with the methodology for successive random additions to generate a fbm k interpolation realization in the vertical cross section between two wells the results appeared realistic and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media copyright c 1993 american geophysical union

diffusion in one dimensional multifractal porous media we examine the scaling properties of one dimensional random walks on media with multifractal diffusivities which is a simple model for transport in scaling porous media we find both theoretically and numerically that the anomalous scaling exponent of the walk is dw 2 k 1 where k 1 is the scaling exponent of the reciprocal spatially averaged dressed resistance to diffusion since k 1 0 the walk is subdiffusive the walkers are effectively trapped in a hierarchy of barriers the trapping is dominated by contributions from a specific order of singularity associated with a phase transition between anomalous and normal diffusion we discuss the implications for transport in porous media

multiscaling analysis and random cascade modeling of dye infiltration 1 we aimed at investigating whether the spatial variability of infiltration in field soils as visualized through dye infiltration experiments is characterized by a multiscaling behavior digitized high resolution dye images from three sites in an experimental catchment in tunisia were analyzed using three indicators of scaling empirical probability distribution functions power spectra and raw statistical moments the two former indicators suggested a general scaling behavior of the data which through the moments analysis was found to be of multiscaling type random cascade processes are frequently used to model multiscaling processes and we fitted the universal multifractal um model of schertzer and lovejoy 1987 to our data the um model closely reproduced the empirical k q functions and simulated fields reproduced key features in the observed ones the results indicate that multiscaling random cascade modeling is useful for statistically describing flow processes and solute transport under field conditions

emergent behavior of a cucker smale type particle model with nonlinear velocity couplings in this note we present a cucker smale type flocking model with nonlinear velocity couplings and derive sufficient conditions for the formation of flocking in terms of communication weight and initial spatial velocity standard deviations

multifractal analysis of flow velocity simulated with the lattice model approach in idealized three dimensional porous media 1 the strange attractor formalism has been used to check whether the flow velocity in idealized three dimensional porous media could be described by the multifractal approach the flow velocity magnitude was simulated with a lattice model in six media with different porosities the results obtained from the analysis of the mass exponent functions and the multifractal and generalized dimensions spectra show that the velocity of the simulated flows has a multifractal type of scaling that becomes stronger as the porosity decreases a correlation analysis has been performed showing the relationships between the statistical properties and the multifractal parameters of the velocity magnitude distribution

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

characterization of permeability anisotropy using wavelet analysis 1 wavelet analysis is an image analysis technique that can extract local information at multiple scales because of this capability wavelet analysis can be used to identify dominant scales in statistically heterogeneous and anisotropic random fields we develop and test a wavelet analysis method for identifying dominant scales and orientations in permeability fields and for identifying boundaries between regions with different dominant orientations we evaluate three different wavelets fully anisotropic morlet wavelet mexican hat wavelet and cauchy wavelet and show that the morlet wavelet is the most effective of these three wavelets in identifying dominant orientations we also investigate the use of several different quantitative wavelet measures in identifying dominant scales and orientations in permeability fields the technique is demonstrated using both a synthetic data set with known characteristics and a laboratory collected permeability data set from massillon sandstone

second order directional derviatives in parametric optimization problems we characterize the second order directional derivative in the sense of ben tal and zowe of a marginal function of the following general form x inf f x y y x this requires us to introduce some geometrical notions as for example the first and second order tangent directions to a multifunction the main results are then applied to some particular types of marginal functions

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

distributed nonlinear control of mobile autonomous multi agents abstract this paper studies the distributed nonlinear control of mobile autonomous agents with variable and directed topology a new distributed nonlinear design scheme is presented for multi agent systems modeled by double integrators with the new design the outputs of the controlled agents asymptotically converge to each other as long as a mild connectivity condition is satisfied moreover the velocity derivative of the output of each agent can be restricted to be within any specified neighborhood of the origin which is of practical interest for systems under such physical constraint the new design is still valid if one of the agents is a leader and the control objective is to achieve leader following as an illustration of the generality and effectiveness of the presented methodology the formation control of a group of unicycle mobile robots with nonholonomic constraints is revisited instead of assuming the point robot model the unicycle model is transformed into two double integrators by dynamic feedback linearization and the proposed distributed nonlinear design method is used to overcome the singularity problem caused by the nonholonomic constraint by properly restricting the velocities simulation results are included to illustrate the theoretical results

stochastic interpolation of aquifer properties using fractional levy motion this paper describes a new technique for conditional simulation of aquifer properties in unobserved regions the technique utilizes a recently introduced model for heterogeneity based on the levy stable family of probability distributions to achieve a higher degree of realism than is possible with gaussian based techniques specifically permeability and porosity variations are modeled using fractional levy motion a new algorithm for producing fractional levy motion conditioned to match known data is introduced two dimensional simulations using actual permeability measurements as input demonstrate that this algorithm produces random fields with the specified properties simulations using a subset of a detailed permeability map of an outcrop as input illustrate how the new method can be used to estimate important flow and transport properties and to quantify uncertainties in these estimates

second order consensus of multi agent systems under limited interaction ranges this paper studies a distributed linear consensus protocol for second order multi agent systems under limited agent interaction ranges in particular two agents can interact with each other only if their distance is within a certain range under a linear consensus protocol with the relative state feedback we derive a sufficient condition on the initial states and the interaction range to achieve the second order consensus by using a lyapunov functional approach finally simulation examples are included to validate the theoretical result

anisotropy lacunarity and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms it has been shown by di federico and neuman 1997 1998a b that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework which views hydraulic conductivity as a random fractal characterized by a power variogram such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments di federico and neuman 1997 have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated fields modes that are homogeneous and isotropic we show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic provided the ratios between principal integral spatial correlation scales are the same for all modes we then analyze the effect of filtering out truncating modes of low high and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous the integral scales of the lowest and highest frequency cutoff modes are related to length scales of the sampling window domain and data support sample volume respectively taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale in a manner previously shown to be consistent with observations in the isotropic case it also allows in principle bridging across scales at a specific locale as well as among locales by adopting either site specific or generalized variogram parameters the introduction of intermediate cutoffs allows us to account in a straightforward manner for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales for example where textural and structural features are associated with distinct ranges of scale such as fractures having discrete ranges of trace length and density which dissect the rock into matrix blocks having corresponding ranges of sizes we explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale variance covariance and spectra of a truncated fractal field we then develop an expression for the equivalent hydraulic conductivity of a box shaped porous block embedded within a multiscale log hydraulic conductivity field under mean uniform flow the block is larger than the support scale of the field but is smaller than a surrounding sampling window consequently its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function conditioned on a known mean value of support scale conductivity across the window are given explicitly by our multiscale theory

multifractal analyses of hydraulic conductivity distributions the concept of a universal multifractal a generalization of a monofractal is a recently developed scaling model for natural phenomena characterized by irregularity presented herein is an effort to use universal multifractal concepts to deal with spatial variations of hydraulic conductivity k which have a significant effect on contaminant transport in the subsurface structure function analyses of four k data sets show that some vertical variations of k display multifractal structures while others are consistent with monofractal behavior in order to make multifractal concepts more useful multifractal noise is introduced and defined as the increments of a multifractal it is concluded that the multifractal formalism of schertzer and lovejoy 1987 has provided a rather general approach for modeling in k variations in the vertical with the exception of horizontal variations of the borden data all results fell within the domain of universal multifractal behavior which includes the monofractal case parameters were well defined in an empirical sense and easy to calculate indicating a robust formalism results were consistent with the recent finding of liu and molz 1997 also non gaussian and scale variant behavior in hydraulic conductivity distributions submitted to water resources research 1997 hereinafter referred to as submitted paper that k variations display increasing heterogeneity at decreasing scales

variational inclusions for contingent derivative of set valued maps in this paper we give two versions of ky fan s inequality for set valued maps acting between normed vector spaces and we consider sufficient conditions to solve a variational inclusion problem concerning derivatives of set valued maps a selection result for set valued maps between finite dimensional vector spaces and its contingent derivative is obtained as well from this result we derive some conditions for the existence of a solution of a generalized variational inequality problem

wijsman convergence of convex sets under renorming let x p be a metric space a natural way to define convergence for a sequence of nonempty closed subsets a of x to a nonempty closed subset a is to insist that we have pointwise convergence of the associated sequence of distance functionals for the sets a to the distance functional for the set a for each x e x c x a lim d x a convergence in this sense is just convergence with respect to the weak topology on the nonempty closed subsets cl x of x determined by the family of distance functionals d x e x e x the so called wijsman topology z l 8 determined by the metric p here of course each functional of the form d x 0 is a function of a closed set variable unfortunately different metrics may give rise to different wijsman topologies even if the metrics are uniformly equivalent see e g 4 section 41 it turns out 6 theorem 3 11 that if we vary our metrics over the class c x of metrics that give rise to the topology of x as well as varying x over x the resulting family

existence and differentiability of metric projections in hilbert spaces this paper considers metric projections onto a closed subset s of a hilbert space if the set s is convex then it is well known that the corresponding metric projections always exist unique and directionally differentiable at boundary points of s these properties of metric projections are considered for possibly nonconvex sets s in particular existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as nearly convex sets

calculus rules on a new set valued second order derivative for convex functions etant donnee une fonction convexe f qui a ete construite a partir d autres fonctions f i dont les differentielles secondes sont aisees a determiner on cherche a exprimer les differentielles secondes de f en fonction des 2 f i

brief paper some necessary and sufficient conditions for second order consensus in multi agent dynamical systems this paper studies some necessary and sufficient conditions for second order consensus in multi agent dynamical systems first basic theoretical analysis is carried out for the case where for each agent the second order dynamics are governed by the position and velocity terms and the asymptotic velocity is constant a necessary and sufficient condition is given to ensure second order consensus and it is found that both the real and imaginary parts of the eigenvalues of the laplacian matrix of the corresponding network play key roles in reaching consensus based on this result a second order consensus algorithm is derived for the multi agent system facing communication delays a necessary and sufficient condition is provided which shows that consensus can be achieved in a multi agent system whose network topology contains a directed spanning tree if and only if the time delay is less than a critical value finally simulation examples are given to verify the theoretical analysis

evidence for non gaussian scaling behavior in heterogeneous sedimentary formations vertical and horizontal fluctuations in permeability and porosity in sedimentary formations are analyzed and are found to be consistent with scaling models based on levy stable probability distributions the approach avoids the assumption of gaussian behavior and is supported by evidence from horizontal and vertical well logs and from permeability measurements on a sandstone outcrop and segments of core from a heterogeneous formation the incremental values in these measurement sequences are accurately modeled as having levy stable distributions the width of the distribution of increments depends on the spatial scale in a manner consistent with scaling behavior the width of the distribution is smaller for horizontal increments than for vertical increments reflecting the reduced variability in the horizontal direction the scaling parameters are in the range associated with antipersistence and are roughly the same magnitude in the vertical and horizontal directions the relationships between different physical properties are briefly studied and it is suggested that they be quantified through off diagonal terms in a multivariate levy width matrix simulations designed to reproduce the observed statistical features are also described these results have some fundamental implications as levy stable distributions require a different set of statistical tools and theoretical methods compared to finite variance distributions

directional differentiability of metric projections onto moving sets at boundary points soient x y des espaces normes de dimension finie et x y une multifonction assignant a chaque xex un ensemble ferme x cy a x sont associees la fonction distance d x y r et la projection metrique a valeur d ensemble x y y on etudie les proprietes differentielles de au voisinage d un point x 0 y 0 avec y 0 point limite de l ensemble x 0

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

consensus condition for linear multi agent systems over randomly switching topologies abstract this paper studies both continuous and discrete time consensus problems for multi agent systems with linear time invariant agent dynamics over randomly switching topologies the switching is governed by a time homogeneous markov process whose state corresponds to a possible interaction topology among agents necessary and sufficient conditions are derived for achieving consensus under a common control protocol respectively it is shown that the effect of switching topologies on consensus is determined by the union of topologies associated with the positive recurrent states of the markov process moreover the effect of random link failures on discrete time consensus is investigated the implications and relationships with the existing results are discussed finally the theoretical results are validated via simulations

ensemble solute transport in two dimensional operator scaling random fields 1 motivated by field measurements of aquifer hydraulic conductivity k recent techniques were developed to construct anisotropic fractal random fields in which the scaling or self similarity parameter varies with direction and is defined by a matrix ensemble numerical results are analyzed for solute transport through these two dimensional operator scaling fractional brownian motion ln k fields both the longitudinal and transverse hurst coefficients as well as the radius of isotropy are important to both plume growth rates and the timing and duration of breakthrough it is possible to create operator scaling fractional brownian motion fields that have more continuity or stratification in the direction of transport the effects on a conservative solute plume are continually faster than fickian growth rates highly non gaussian shapes and a heavier tail early in the breakthrough curve contrary to some analytic stochastic theories for monofractal k fields the plume growth rates never exceed a mercado s 1967 purely stratified aquifer growth rate of plume apparent dispersivity proportional to mean distance apparent superstratified growth must be the result of other demonstrable factors such as initial plume size

discrimination of fractional brownian movement and fractional gaussian noise structures in permeability and related property distributions with range analyses recent studies have shown that a useful scheme to characterize subsurface heterogeneity is to use fractional brownian motion fbm and fractional gaussian noise fgn commonly used techniques to detect these fractal distributions include rescaled range spectral and semivariogram analyses although rescaled range analysis is considered to be superior to spectral or semivariogram techniques it cannot be used to clearly discriminate fgn and fbm from a given data set in some cases on the basis of the fact that both adjusted range and rescaled range analyses for a true fgn data set should yield the same slope value in the relevant log log plots a simple scheme is developed to discriminate fgn and fbm using a combination of the two analyses the usefulness of the proposed scheme is evaluated with one dimensional computational experiments and by an application to a data set of observed air permeability logs lack of clear discrimination may have contributed to inconsistent past results and it is recommended that old data sets be reanalyzed

a general fractal model of flow and solute transport in randomly heterogeneous porous media 1 one of the challenges of hydrology is modeling flow and solute transport through media that are not uniformly porous this paper proposes a general fractal model of flow and solute transport in randomly heterogeneous porous media we describe the random field using fractional levy motion flm which more effectively represents a field with a high degree of variability understanding flow and solute transport in the flm field expands the applicability of stochastic theory following the work of v di federico and s p neuman 1997 and v di federico et al 1999 we derive truncated isotropic and anisotropic power variograms for the flm fields the results are analogous to but more general than those of fractional brownian motion fbm v di federico and s p neuman 1997 and v di federico et al 1999 for a random field when the levy index in flm is replaced by a constant of 2 fbm is recovered this study employs first order perturbation analyses of flow and solute transport in two dimensional flm fields for the stationary fbm and flm fields macrodispersion coefficients are compared the plume is found to grow slowest in the flm field but will show highest concentration the proposed fractal model is general in the sense that when the levy index of flm equals 2 the results of flow and solute transport for the commonly used fbm model are recovered when the upper cutoff of the truncated power variogram is close to the lower cutoff the results for the stationary exponential or gaussian model can be well approximated since the proposed general fractal model has broader applications than the stationary and fbm models it simulates flow and solute transport in more different scenarios to yield more accurate modeling results

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control