measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

an efficient algorithm for solving integro differential equations system in this paper an application of he s homotopy perturbation hpm method is applied to solve of system of integro differential equations the results reveal that the hpm is very effective and simple

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

adomian decomposition method for solving the volterra integral form of the lane emden equations with initial values and boundary conditions in this paper we use the systematic adomian decomposition method to handle the integral form of the lane emden equations with initial values and boundary conditions the volterra integral form of the lane emden equation overcomes the singular behavior at the origin x 0 we confirm our belief that the adomian decomposition method provides efficient algorithm for analytic approximate solutions of the equation our results are supported by investigating several numerical examples that include initial value problems and boundary value problems as well finally we consider the modified decomposition method of rach adomian and meyers for the volterra integral form

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 paper necessary conditions for hadamard factorizations of hurwitz polynomials a property of hurwitz polynomials is related with the hadamard product garloff and wagner proved that hadamard products of hurwitz polynomials are hurwitz polynomials and garloff and shrinivasan shown that there are hurwitz polynomials of degree 4 which do not have a hadamard factorization into two hurwitz polynomials of the same degree 4 in this paper we give necessary conditions for an even degree hurwitz polynomial to have a hadamard factorization into two even degree hurwitz polynomials such conditions are given in terms of the coefficients of the given polynomial alone furthermore we show that if an odd degree hurwitz polynomial has a hadamard factorization then a system of nonlinear inequalities has at least one solution

the variational iteration method a highly promising method for solving the system of integro differential equations this paper applies he s variational iteration method for solving two systems of volterra integro differential equations the solution process is illustrated and various physically relevant results are obtained comparison of the obtained results with exact solutions shows that the used method is an effective and highly promising method for various classes of both linear and nonlinear integro differential equations

a comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow this paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity the ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant seven geostatistical methods fast fourier transform ff fractal simulation fs linearized cokriging lc linearized semianalytical ls maximum likelihood ml pilot point pp and sequential self calibration ss were compared on four synthetic data sets each data set had specific features meeting or not classical assumptions about stationarity amenability to a geostatistical description etc the comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km four of the methods ls ml pp and ss were identified as being approximately equivalent for the specific problems considered the magnitude of the variance of the transmissivity fields which went as high as 10 times the generally accepted range for linearized approaches was not a problem for the linearized methods when applied to stationary fields that is their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods nonstationarity of the true transmissivity field or the presence of anomalies such as high permeability fracture zones was however more of a problem for the linearized methods the importance of the proper selection of the semivariogram of the log10 t field or the ability of the method to optimize this variogram iteratively was found to have a significant impact on the accuracy and precision of the travel time predictions use of additional transient information from pumping tests did not result in major changes in the outcome while the methods differ in their underlying theory and the codes developed to implement the theories were limited to varying degrees the most important factor for achieving a successful solution was the time and experience devoted by the user of the method

propagation and stability of waves of electrical activity in the cerebral cortex nonlinear equations are introduced to model the behavior of the waves of cortical electrical activity that are responsible for signals observed in electroencephalography these equations incorporate nonlinearities axonal and dendritic lags excitatory and inhibitory neuronal populations and the two dimensional nature of the cortex while rendering nonlinear features far more tractable than previous formulations both analytically and numerically the model equations are first used to calculate steady state levels of cortical activity for various levels of stimulation dispersion equations for linear waves are then derived analytically and an analytic expression is found for the linear stability boundary beyond which a seizure will occur the effects of boundary conditions in determining global eigenmodes are also studied in various geometries and the corresponding eigenfrequencies are found numerical results confirm the analytic ones which are also found to reproduce existing results in the relevant limits thereby elucidating the limits of validity of previous approximations

brief paper smooth trivial vector bundle structure of the space of hurwitz polynomials due to the importance of the robustness of stable systems the properties of the set of hurwitz real polynomials are interesting to study among these properties is that the set is an open and not a convex set and recently it was proved that the set of hurwitz polynomials of degree n with positive coefficients h n is contractible now we will prove that the set h n is a smooth trivial vector bundle over h n k of rank k

geostatistical inference of hydraulic conductivity and dispersivities from hydraulic heads and tracer data 1 in groundwater hydraulic heads and solute arrival times depend primarily on the hydraulic conductivity field and hydraulic boundary conditions the spread of breakthrough curves in contrast depends also on longitudinal and transverse dispersion coefficients the shape of point like measured breakthrough curves can be reproduced by simulations only when appropriate dispersivities are applied the values and spatial distributions of dispersivities depend on the resolution of the underlying hydraulic conductivity field we present a geostatistical method for the joint estimation of log conductivity and log dispersivities from measurements of hydraulic heads and temporal moments of local breakthrough curves the parameter fields are considered random space variables and they are conditioned on the measurements by bayesian inference the estimated longitudinal and transverse dispersivities are to be applied in conjunction with the estimated conductivity field we apply our technique to data of a technical scale tracer experiment in the particular application the amount and quality of measured data are sufficient to infer the conductivity distribution at a spatial resolution at which the spread of locally obtained breakthrough curves is dominated by pore scale transverse dispersion

a higher order finite difference method for solving a system of integro differential equations abstract a finite difference method and gauss method of numerical integration are used to approximate the integro differential system of eq 1 with the global error z h o h4 the resulting system of algebraic equations is solved by efficient implicit iterations a mathematica module is designed for the purpose of testing and using the method an example of an integral differential equation with the green s function as the kernel is solved

local lyapunov exponents of the quasi geostrophic ocean dynamics the predictability of the quasi geostrophic ocean model is considered in this paper this is a simple dynamical model which assumes the ocean depth to be divided into n layers of different water density this model is driven by the wind stress on the surface it includes effects of lateral and bottom friction of the earth rotation and the nonlinear interaction between adjacent layers internal instability of the system leads to the rapid divergence of its trajectories limiting time of deterministic prediction of the system to estimate the divergence rate of trajectories we compute lyapunov exponents of this system along with lyapunov exponents their generalization local lyapunov exponents are computed as a measure of divergence rate on finite time scales these exponents provide us with the information about principal mechanisms of local instability they can be useful in the forecasting of the forecast skill in meteorology and oceanology numerous studies are performed in order to understand the influence of physical form of the basin number of layers and numerical grid resolution parameters on the predictability of the system to study the geographical structure of the perturbation the distribution of initial error with the fastest norm growth is distinguished this distribution shows us the initial data structure the most sensible to errors and points out the regions where the most accuracy should be required the structure of this function depends much on the time scale of prediction

dispersion on kriged hydraulic conductivity fields 1 geostatistical interpolation of log conductivity measurements leads to velocity fields lacking small scale variability whereas large scale features are covered well the unresolved small scale velocity fluctuations cause underestimation of macrodispersion dilution and mixing in subsequent transport calculations the same holds when the best linear unbiased estimate of geostatistical inverse methods is used in flow and transport simulations considering a regular grid of measurement points we approximate the lack of dispersion to which we refer as correction dispersion by applying first order theory to the conditional covariance of the kriged log conductivity field the nonstationarity of the conditional covariance is removed by averaging it over space we distinguish between macrodispersion describing the rate of change in the second spatial moments of a large plume and effective dispersion of a plume introduced as a point source parameterizing dilution and mixing depending on the objective we either add a correction macrodispersion tensor or a correction effective dispersion tensor to the true local dispersion tensor in transport calculations on the smoothed log conductivity field the numerical application to a two dimensional periodic domain shows excellent agreement in the one and two particle moments between simulations on the highly resolved field and those on the interpolated one using corrected values for the local dispersion coefficients

a comparison of two numerical methods for integrating a quasi geostrophic multilayer model of ocean circulations to study the feasibility of using the finite elements method for ocean modelling a quasi geostrophic eddy resolving general circulation model has been driven using either a classical finite differences technique or a finite elements technique this feasibility has been clearly demonstrated and a comparison of the models performances shows that for simulations of realistic basins the finite elements model cpu cost is only four times the finite difference model cost

a reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

a simple algorithm for exact solutions of systems of linear and nonlinear integro differential equations a simple algorithm is used to achieve exact solutions of systems of linear and nonlinear integro differential equations arising in many scientific and engineering applications the algorithm does not need to find the adomain polynomials to overcome the nonlinear terms in adomain decomposition method adm it does not need to create a homotopy with an embedding parameter as in homotopy perturbation method hpm and optimal homotopy asymptotic method oham unlike vim it does not need to find lagrange multiplier in this manuscript no restrictive assumptions are taken for nonlinear terms the applied algorithm consists of a single series in which the unknown constants are determined by the simple means described in the manuscript the outcomes gained by this algorithm are in excellent concurrence with the exact solution and hence proved that this algorithm is effective and easy four systems of linear and nonlinear integro differential equations are solved to prove the above claims and the outcomes are compared with the exact solutions as well as with the outcomes gained by already existing methods

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

solving linear integro differential equations system by using rationalized haar functions method in this paper we use rationalized haar functions to estimate the solution of linear integro differential equations system properties of rationalized haar functions are first presented and the operational matrix of the product of two rationalized haar functions vector is utilized to reduce the computation of linear integro differential equations system to some algebraic equations by using numerical examples we show our estimation have a good degree of accuracy

highly parameterized inverse estimation of hydraulic conductivity and porosity in a three dimensional heterogeneous transport experiment 1 assessing the impact of parameter estimation accuracy in models of heterogeneous three dimensional 3 d groundwater systems is critical for predictions of solute transport a unique experimental data set provides concentration breakthrough curves btcs measured at a 0 253 cm3 scale over the 13 8 8 cm3 domain 53 000 measurement locations advective transport is used to match the first temporal moments of btcs or mean arrival times m1 averaged at 0 253 and 1 0 cm3 scales through simultaneous inversion of highly parameterized heterogeneous hydraulic conductivity k and porosity fields pilot points parameterize the fields within eight layers of the 3 d medium and estimations are completed with six different models of the k relationship parameter estimation through advective transport shows accurate estimation of the observed m1 values results across the six different k relationships have statistically similar fits to the observed m1 values and similar spatial estimates of m1 along the main flow direction the resulting fields provide the basis for forward transport modeling of the advection dispersion equation ade using the estimated k and fields demonstrates that advective transport coupled with inversion using dense spatial field parameterization provides an efficient surrogate for the ade these results indicate that there is not a single set of model parameters or a single k relationship that leads to a best representation of the actual experimental sand packing pattern i e nonuniqueness additionally knowledge of the individual sand k and values along with their arrangement in the 3 d experiment does not reproduce the observed transport results at small scales small scale variation in the packing and mixing of the sands causes large deviations from the expected transport results as highlighted in forward ade simulations highly parameterized inverse estimation is able to identify those regions where variations in mixing and packing alter the expected property values and significantly improve results relative to the naive application of the experimentally derived property values impacts of the observation scale the scale over which results are averaged and the number of observations and parameters on the final estimations are also examined results indicate existence of a representative element volume rev at 0 25 cm3 the existence of subgrid scale heterogeneity that impacts transport and the accuracy of highly parameterized models with even relatively small amounts of observations finally this work suggests that local heterogeneity features below the rev scale are difficult to incorporate into parameterized models highlighting the importance of addressing prediction uncertainty for small scale variability i e uncaptured variability in modeling practice