stochastic analysis of dispersion in unsteady flow in heterogeneous aquifers the dispersion of a solute plume resulting from unsteady flow in three dimensional heterogeneous porous media was analyzed using stochastic continuum theory asymptotic stochastic solutions of the perturbed unsteady flow and solute transport equations were 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 and an input covariance describing the temporal variability of the mean hydraulic gradient the flow equation was approximated by neglecting the influence of internal storage due to medium and fluid compressibility specific storage ss 0 the predictive expression for the macrodispersivity tensor is the sum of two components a spatial variability component identical to previous steady state theory and an unsteady component two special cases of unsteady flow were examined variation only in the magnitude and variation only in the direction of the hydraulic gradient gradient magnitude variation produces a slightly larger longitudinal macrodispersivity than does the steady flow case whereas gradient direction variation produces a significantly larger transverse macrodispersivity longitudinal and horizontal transverse macrodispersivities predicted with the unsteady stochastic theory were shown to be of a magnitude similar to observed values from the borden cape cod and columbus tracer experiments

philos type oscillation criteria for linear differential equations with impulsive effects abstract the present paper deals with the problem of oscillation for a second order linear differential equation in which an impulsive effect is considered this equation can be regarded as an equation of motion in which the moving speed of a mass point changes sharply by some influence it is proved that there is a case that the mass point may oscillate due to the influence of the impulsive effect even if the mass point does not oscillate in the model removing the impulsive effect it is also shown that the obtained results extend some previous ones through the use of an example

oscillation criteria for second order quasilinear functional differential equations abstract some oscillation criteria for the second order quasilinear functional differential equations are obtained our results generalize and improve some known results of both differential equations and delay differential equations

high order hamiltonian splitting for the vlasov poisson equations we consider the vlasov poisson equation in a hamiltonian framework and derive new time splitting methods based on the decomposition of the hamiltonian functional between the kinetic and electric energy assuming smoothness of the solutions we study the order conditions of such methods it appears that these conditions are of runge kutta nystrom type in the one dimensional case the order conditions can be further simplified and efficient methods of order 6 with a reduced number of stages can be constructed in the general case high order methods can also be constructed using explicit computations of commutators numerical results are performed and show the benefit of using high order splitting schemes in that context complete and self contained proofs of convergence results and rigorous error estimates are also given

higher order volume preserving schemes for charged particle dynamics a class of higher order numerical methods for advancing the charged particles in a general electromagnetic field is developed based on processing technique by taking the volume preserving methods as the kernel the processed methods are still volume preserving and preserve the conservative quantities for the lorenz force system moreover this class of numerical methods are explicit and are more efficient compared with other higher order composition methods linear stability analysis is given by applying the numerical methods to the test equation it is shown that the newly constructed higher order methods have the better stability property this allows the use of larger step sizes in their implementation

a note on conservative transport in anisotropic heterogeneous porous media in the presence of small amplitude transients the late time macrodispersion coefficients are obtained for the case of flow in the presence of a small scale deterministic transient in a three dimensional anisotropic heterogeneous medium the transient is assumed to affect only the velocity component transverse to the mean flow direction and to take the form of a periodic function for the case of a highly stratified medium these late time macrodispersion coefficients behave largely as the standard coefficients used in the transport equation only in the event that the medium is isotropic is it probable that significant deviations from the standard coefficients would occur

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

composition constants for raising the orders of unconventional schemes for ordinary differential equations many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general purpose software designed to solve numerically a wide range of differential equations if those properties are to be exploited fully for the sake of better numerical stability accuracy and or speed the differential equations may have to be solved by unconventional methods this short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods called reflexive

lagrangian analysis of transport in heterogeneous formations under transient flow conditions the coupled effects of porous media heterogeneity and flow unsteadiness on transport are investigated we address the problem using a stochastic lagrangian approach the log condictivity is modeled as a space random function however the flow unsteadiness is considered as deterministic and it affects the magnitude and direction of the mean head gradient using a first order solution in terms of the log conductivity variance we developed a three dimensional solution that includes the velocity covariances the macrodispersion coefficients and the displacement tensor all as a function of the media heterogeneity and the unsteady mean flow parameters the model is applied to the case of a periodic variation in the flow direction we show that the effects of unsteadiness on the longitudinal spread are insignificant while it affects strongly the transverse spread in the form of an added constant dispersivity results obtained in previous studies are discussed and are shown to be particular cases of the present results

oscillation criteria for certain nonlinear differential equations we present new oscillation criteria for certain classes of second order nonlinear differential equations and delay differential equations obtained by using an integral averaging technique our theorems complement a number of existing results and handle the cases which are not covered by known criteria

construction of high order force gradient algorithms for integration of motion in classical and quantum systems a consequent approach is proposed to construct symplectic force gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations within this approach the basic algorithms are first derived up to the eighth order by direct decompositions of exponential propagators and further collected using an advanced composition scheme to obtain the algorithms of higher orders contrary to the scheme proposed by chin and kidwell phys rev e 62 8746 2000 where high order algorithms are introduced by standard iterations of a force gradient integrator of order four the present method allows one to reduce the total number of expensive force and its gradient evaluations to a minimum at the same time the precision of the integration increases significantly especially with increasing the order of the generated schemes the algorithms are tested in molecular dynamics and celestial mechanics simulations it is shown in particular that the efficiency of the advanced fourth order based algorithms is better approximately in factors 5 to 1000 for orders 4 to 12 respectively the results corresponding to sixth and eighth order based composition schemes are also presented up to the sixteenth order for orders 14 and 16 such highly precise schemes at considerably smaller computational costs allow to reduce unphysical deviations in the total energy up in 100000 times with respect to those of the standard fourth order based iteration approach

sturmian comparison theorem for half linear second order differential equations let be defined by s s p 2 s with p 1 a fixed number we extend sturm comparison theorem of the linear differential equation to the nonlinear differential equation by using the wirtinger inequality a lyapunov inequality and some oscillation criteria of e are also given

interval criteria for oscillation of second order functional differential equations by using averaging functions new interval oscillation criteria are established for the second order functional differential equation r t x t a 1x t f t x t x t t x t x t t 0 t t 0that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of t 0 rather than on the whole half line our results can be applied to three cases ordinary delay and advance differential equations in the case of half linear functional differential equations our criteria implies that the t t t delay and gt t t advance cases do not affect the oscillation in particular several examples are given to illustrate the importance of our results

free lie algebras generalized witt formula and the denominator identity abstract let be a countable abelian semigroup satisfying a suitable finiteness condition and let l l be the free lie algebra generated by a graded vector space v over c in this paper from the denominator identity we derive a dimension formula for the homogeneous subspaces of the free lie algebra l l and discuss numerous applications of our dimension formula to various interesting cases our dimension formula will be expressed in terms of the witt partition functions various expressions of the witt partition functions will give rise to a number of interesting combinatorial identities as a special case we obtain a recursive relation for the coefficients of the elliptic modular function j

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

interval oscillation criteria for second order linear differential equations with impulsive effects abstract this paper discusses how the oscillation of solutions changes by adding an impulsive term to a second order linear differential equation having only a restoring force term but no damping term due to the effect of this impulsive term the moving speed of a mass point of the equation of motion described by the second order linear differential equation changes discontinuously oscillation theorems are given for this impulsive differential equation as is well known when the restoring force term is small all nontrivial solutions of the equation with no impulsive effects do not oscillate even when the restoring force term is small if the action of the impulsive term compensates the oscillation criteria obtained are satisfied and all nontrivial solutions oscillate several examples are given to confirm this fact and some figures are shown to depict the solution curves of these examples as shown by the obtained oscillation theorems there are cases that the impulsive effect promotes the oscillation of solutions but on the contrary it may suppress the oscillation of solutions it is shown by a simulation that there is a case where the impulsive effect suppress the oscillation of solutions finally the obtained results are compared with the previous ones

on the numerical integration of ordinary differential equations by processed methods we provide a theoretical analysis of the processing technique for the numerical integration of odes we get the effective order conditions for processed methods in a general setting so that the results obtained can be applied to different types of numerical integrators we also propose a procedure to approximate the postprocessor such that its evaluation is virtually cost free the analysis is illustrated for a particular class of composition methods

general theory of fractal path integrals with applications to many body theories and statistical physics a general scheme of fractal decomposition of exponential operators is presented in any order m namely exp x a b sm x o xm 1 for any positive integer m where sm x et1a et2b et3a et4b etma with finite m depending on m a general recursive scheme of construction of tj is given explicitly it is proven that some of tj should be negative for m 3 and for any finite m nonexistence theorem of positive decomposition general systematic decomposition criterions based on a new type of time ordering are also formulated the decomposition exp x a b sm x n n o xm 1 nm yields a new efficient approach to quantum monte carlo simulations

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

symplectic integration with processing a general study the number of conditions to be satisfied by the operators k and p in symplectic integrators with processing given by epe hke p is determined for a hamiltonian of the form h cal a cal b the conditions for k are explicitly written up to order six and used to obtain more efficient methods with fewer evaluations per step than other symplectic integrators special cases in which the number of conditions for the kernel is drastically reduced are also studied it is shown that the kernel completely determines the optimal method one can obtain by processing

oscillatory behavior of solutions of certain second order nonlinear differential equations abstract in this paper some simple inequalities are used to offer sufficient conditions for the oscillation of all solutions of the differential equation formula where 0 p q with p q odd integers or p even and q odd integers several examples which dwell upon the importance of our results are also included

composition methods for differential equations with processing we construct numerical integrators for differential equations up to order 12 obtained by composition of basic integrators the following cases are considered i composition for a system separable in two solvable parts ii composition using as basic methods a first order integrator and its adjoint iii composition using second order symmetric methods and iv composition using fourth order symmetric methods each scheme is implemented with a processor or corrector to improve their efficiency and this can be done virtually cost free

quantum monte carlo methods and general decomposition theory of exponential operators and symplectic integrators abstract the present paper reviews briefly the basic ideas of the quantum monte carlo methods the general decomposition theory of ordered exponentials and symplectic integrators

explicit symplectic integrators using hessian vector products in 1991 rowlands proposed an effectively fourth order effectively two stage explicit symplectic integrator based on using a hessian vector product to modify the force evaluation in the leapfrog method and evidence indicates that for modest accuracy this method is highly competitive here we explore the possible existence of even more efficient fourth order explicit symplectic integrators also based on the use of hessian vector products and the concept of effective order first it is shown that the cost of a force evaluation plus a hessian vector product is less than twice the cost of the force alone for a sum of two body interactions then a new method is found that is generally better than both the method of rowlands and that of calvo according to both a theoretical measure of the error and limited numerical experiments the basic motivation behind the new method is quite simple do a hessian vector computation only every other step significantly cutting costs while only marginally increasing the error the idea of effective order means that we allow for both the possibility of preprocessing the initial values before application of the basic method and the possibility of postprocessing the values obtained by the basic method but at output points only for some applications processing is unnecessary but in any case processing has been shown to be possible at low additional cost the derivation of the new method illustrates how to simplify by the use of h series the determination of parameters for methods of increased effective accuracy