the eigenvalues of second order spectral differentiation matrices the eigenvalues of the pseudospectral second derivative matrix with homogeneous dirichlet boundary conditions are important in many applications of spectral methods this paper investigates some of their properties numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately but the errors in the remaining ones are large it is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid in particular it is proved that 7r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a chebyshev distribution of nodes and six points per wavelength for a uniform distribution these results are in agreement with the observed fractions of accurate eigenvalues by using the characteristic polynomial a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2 or better the effect of filtering on the eigenvalues is studied numerically

stallings foldings and subgroups of free groups we re cast in a more combinatorial and computational form the topological approach of j stallings to the study of subgroups of free groups

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

recurrence transience and bounded harmonic functions for diffusions in the plane we give conditions for transience and recurrence for certain diffusion processes in the plane which look recurrent in certain sectors and transient in others we also give conditions for the convergence of diffusion paths to paths of the deterministic dynamical system corresponding to the first order part of l the generator of the process this in turn is related to the question of existence of bounded harmonic functions for the operator l conditions are given for the existence and nonexistence of bounded harmonic functions for l

the barycentric weights of rational interpolation with prescribed poles abstract we introduce a method for calculating rational interpolants when some but not necessarily all of their poles are prescribed the algorithm determines the weights in the barycentric representation of the rationals it simply consists in multiplying each interpolated value by a certain number computing the weights of a rational interpolant without poles and finally multiplying the weights by those same numbers the supplementary cost in comparison with interpolation without poles is about v 2 n where v is the number of poles and n the number of interpolation points we also give a condition under which the computed rational interpolation really shows the desired poles

variational principles and lagrangian functions for stochastic processes and their dissipative statistical descriptions abstract a variational lagrangian formulation for stochastic processes and for the evolution equations of the associated probability density functions is developed particular attention is dedicated to poisson kac processes possessing finite propagation velocity the variational formulation in terms of lagrangian and hamiltonian densities permits to address different forms of reversibility characterizing these processes with respect to processes driven by nowhere differentiable wiener fluctuations and associated with the concepts of dynamic and statistical reversibility the latter property i e the statistical reversibility implies that the extended markov operator associated with poisson kac and generalized poisson kac processes forms a group continuously parametrized with respect to time

monotonicity considerations for saturated unsaturated subsurface flow it is demonstrated that monotonicity is a sufficient condition to ensure that no new nonphysical local maxima and minima can be produced in the discrete nonlinear unsaturated flow equation monotonicity conditions are derived for various types of weighting for the mobility term the basic discretization is of finite element type but the results can be extended to finite volume discretizations central weightings are only conditionally monotone while upstream weightings are unconditionally monotone sample computations are given for highly heterogeneous problems

adaptive point shifts in rational approximation with optimized denominator classical rational interpolation is known to suffer from several drawbacks such as unattainable points and randomly located poles for a small number of nodes as well as an erratic behavior of the error as this number grows larger in a former article we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial using the barycentric representation of the interpolants in order to improve upon the condition of the derivatives in the solution of differential equation we have then experimented with a conformal point shift suggested by kosloff and tal ezer as it turned out such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval this leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function such shifts are also constructed for functions with several shocks away from the extremities of the interval

growth of subalgebras and subideals in free lie algebras abstract we investigate subalgebras in free lie algebras the main tool being relative growth and cogrowth functions our study reveals drastic differences in the behavior of proper finitely generated subalgebras and nonzero subideals for instance the growth of a proper finitely generated subalgebra h of a free lie algebra l with respect to any fixed free basis x is exponentially small compared to the growth of the whole of l quite opposite the cogrowth of any nonzero subideal s is exponentially small compared to the growth of l

a probabilistic approach to positive harmonic functions in a slab with alternating dirichlet and neumann boundary conditions let q rd x 1x 1 d 2 be a d 1 dimensional slab denote points z e rd i by z r o y where r 0 e o oo x sd i and y e r denoting the boundary of the slab by p oq let

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

the linear rational collocation method we introduce the collocation method based on linear rational interpolation for solving general hyperbolic problems prove its stability and its convergence in weighted norms and give numerical examples for its use

optimized point shifts and poles in the linear rational pseudospectral method for boundary value problems due to their rapid often exponential convergence as the number n of interpolation collocation points is increased polynomial pseudospectral methods are very efficient in solving smooth boundary value problems however when the solution displays boundary layers and or interior fronts this fast convergence will merely occur with very large n to address this difficulty we present a method which replaces the polynomial ansatz with a rational function r and considers the physical domain as the conformal map g of a computational domain g shifts the interpolation points from their classical position in the computational domain to a problem dependent position in the physical domain starting from a map by bayliss and turkel we have constructed a shift that can in principle accomodate an arbitrary number of fronts its parameters as well as the poles of r are optimized numerical results demonstrate how g best accomodates interior fronts while the poles also handle boundary layers

on monotonicity of difference schemes for computational physics criteria are developed for monotonicity of linear as well as nonlinear difference schemes associated with the numerical analysis of systems of partial differential equations integrodifferential equations etc difference schemes are converted into variational forms that satisfy the boundary maximum principle and also allow the investigation of monotonicity for nonlinear operators using linear patterns sufficient conditions are provided to review the monotonicity of single and coupled difference schemes necessary as well as necessary and sufficient conditions for monotonicity of explicit schemes are also developed the notion of submonotone difference schemes is considered and the associated criteria are developed we discuss the interrelationship between monotonicity submonotonicity and stability some known schemes serve as examples demonstrating the implementation of the developed approaches among these examples we describe the possibility that stable schemes such as total variation diminishing tvd as well as monotonicity preserving can produce spurious oscillations

estimation of finite difference interblock conductivities for simulation of infiltration into initially dry soils simulations of transient unsaturated vertical flow in different initially dry homogeneous soils using richards equation suggest that for a relatively coarse numerical grid there is no universal weighting scheme to estimate interblock conductivities capable of giving an accurate resolution of the resulting steep wetting fronts except for the scheme associated with the kirchhoff transformation kt the principal drawback of kt is that it can hardly be applied to heterogeneous soils a new asymptotic weighting aw scheme based on the asymptotic behavior of the hydraulic conductivity function in the vicinity of residual saturation is proposed numerical tests demonstrate that the aw scheme describes wetting fronts in homogeneous soils nearly as accurately as a scheme based on kt applicability and robustness of the aw scheme for simulating water infiltration into heterogeneous soils having substantial vertical variation of hydraulic properties are also demonstrated extension of the proposed scheme to higher dimensions is straightforward

asymptotic properties of solutions of multidimensional stochastic differential equations letx t r d be the solution of the stochastic equationdx t b x t dt x t dw t wherew t denotes a standard wiener process the aim of the paper is to clarify under which conditions the drift term or the diffusion term is of negligible significance for the long term behaviour ofx t

fluctuations far from equilibrium hyperbolic transport the problem of the statistical properties of fluctuations in systems far from equilibrium is addressed the discussion is based on a variational approach to hyperbolic transport equations built in a space spanned by the usual thermodynamic properties plus a set of potential functions associated with them the enlarged space characterizes the far from equilibrium states of the system and fluctuations in the potential functions satisfy the chapman kolmogorov equation well known results on processes near equilibrium are recovered in the parabolic transport limit

three dimensional simulation of unstable gravity driven infiltration of water into a porous medium infiltration of water in dry porous media is subject to a powerful gravity driven instability although the phenomenon of unstable infiltration is well known its description using continuum mathematical models has posed a significant challenge for several decades the classical model of water flow in the unsaturated flow the richards equation is unable to reproduce the instability here we present a computational study of a model of unsaturated flow in porous media that extends the richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively the extended model is based on a phase field formulation and is fourth order in space the new model poses a set of challenges for numerical discretizations such as resolution of evolving interfaces stiffness in space and time treatment of singularly perturbed equations and discretization of higher order spatial partial differential operators we develop a numerical algorithm based on isogeometric analysis a generalization of the finite element method that permits the use of globally smooth basis functions leading to a simple and efficient discretization of higher order spatial operators in variational form we illustrate the accuracy efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media we simulate for the first time unstable gravity driven infiltration in three dimensions and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux

spectral integration and two point boundary value problems a numerical method for two point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for chebyshev nodes the method is stable achieves superalgebraic convergence and requires o n log n operations where n is the number of nodes in the discretization although stable spectral methods have been constructed in the past they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill conditioning introduced by that process

non standard shocks in the buckley leverett equation abstract it is shown how delta shock waves which consist of dirac delta distributions and classical shocks can be used to construct non monotone solutions of the buckley leverett equation these solutions are interpreted using a recent variational definition of delta shock waves in which the rankine hugoniot deficit is explicitly accounted for 6 the delta shock waves are also limits of approximate solutions constructed using a recent extension of the weak asymptotic method to complex valued approximations 15 finally it is shown how these non standard shocks can be fitted together to construct similarity and traveling wave solutions which are non monotone but still admissible in the sense that characteristics either enter or are parallel to the shock trajectories

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

stability study of a constant volume thin film flow we study the stability of a constant volume of fluid spreading down an incline in contrast to the commonly considered flow characterized by constant fluid flux in the present problem the base flow is time dependent we present a method to carry out consistently linear stability analysis based on simultaneously solving the time evolution of the base flow and of the perturbations the analysis is performed numerically by using a finite difference method supplemented with an integral method developed here the computations show that after a short transient stage imposed perturbations travel with the same velocity as the leading contact line the spectral analysis of the modes evolution shows that their growth rates are in general time dependent the wavelength of maximum amplitude ensuremath lambda max decreases with time until it reaches an asymptotic value which is in good agreement with experimental results we also explore the dependence of ensuremath lambda max on the cross sectional fluid area a and on the inclination angle ensuremath alpha of the substrate for considered small a s corresponding to small bond numbers we find that the dependence of ensuremath lambda max on a is in good agreement with experimental data this dependence differs significantly from the one observed for the films characterized by much larger a s and bond numbers we also predict the dependence of ensuremath lambda max on the inclination angle ensuremath alpha

brownian motion and harmonic functions on rotationally symmetric manifolds we consider brownian motion x on a rotationally symmetric manifold m g mathbb r n ds 2 ds 2 dr 2 g r 2 d theta 2 an integral test is presented which gives a necessary and sufficient condition for the nontriviality of the invariant sigma field of x hence for the existence of nonconstant bounded harmonic functions on m g conditions on the sectional curvatures are given which imply the convergence or the divergence of the test integral

the linear rational pseudospectral method with iteratively optimized poles for two point boundary value problems an algorithm is proposed which improves upon the polynomial pseudospectral method for solving linear two point boundary value problems in the latter the collocation points are the vertical projection onto the interval of points equidistant or nearly equidistant on the circle and they therefore accumulate in the vicinity of the extremities of the interval thus the method is well suited for solving problems whose solutions have boundary layers but not as good at approximating solutions with large gradients shocks away from the extremities of their domain of definition our idea is to modify the polynomial ansatz by attaching a denominator so as to make it a rational interpolant the denominator is then successively optimized in an iterative procedure with each step consisting of the solution of two problems an optimization of the denominator for given values of the approximation to the solution u at the interpolation points and a collocation in the linear space of the rational interpolants with the just obtained fixed denominator to obtain new approximate values of u we show the efficiency of a galerkin version of the method and discuss the power of the collocation version with several numerical examples

the entropy of graded algebras we define the entropy of a graded algebra a n an by limsupn diman this is related to the notion of entropy in symbolic dynamics and could serve as a natural dimension concept for algebras with exponential growth we study the entropy of quotients and subalgebras of free associative algebras and free lie algebras we also study the behavior of the entropy function under free products and obtain several characterizations of free algebras in terms of entropy

on the incompatibility of richards equation and finger like infiltration in unsaturated homogeneous porous media 1 it is demonstrated by means of a mathematical proof that richards equation in principle cannot admit finger like solutions for three dimensional homogeneous unsaturated porous media flow subject to monotone boundary conditions this is demonstrated for any reasonable type of homogeneous porous material the result is not dependent on any particular form of the hydraulic conductivity or the retention curve moreover it is explained why hysteresis of the retention curve does not play any role in the proof consequently the proof is true for any type of hysteretic behavior of the retention curve an alternative approach to finger flow modeling is discussed which uses the ideas of cellular automata

development and evaluation of closed form expressions for hysteretic soil hydraulic properties a concise representation of hysteretic soil hydraulic properties is given based on a combination of m t van genuchten s 1980 parametric k h model and p s scott et al s 1983 empirical hysteresis model modified to account for air entrapment the resulting model yields compact closed form expressions for the hysteretic water retention curve h and soil water capacity c h as well as for the hydraulic conductivity function k h depending on the degree of simplification involved the model entails a total of 6 9 parameters which can be calibrated from direct measurements of h and saturated conductivity or by an inverse solution approach from transient flow experiments comparison of model predicted and measured k h relations for eight soils revealed one case in which model predictions were very poor model accuracy was judged to be acceptably good in the other cases mualern s modified y mualem 1984 dependent domain model was found to be more accurate for soils with very narrow pore size distributions use of a simplified version of the proposed model with two parameters eliminated provided overall accuracy very similar to that of the more complex model numerical simulations of flow during transient infiltration and drainage using the proposed model and a variant of y mualem s 1984 modified dependent domain model did not differ greatly and agreed reasonably well with experimental water content distributions even when scanning curves were not described very accurately by contrast simulations without consideration of hysteresis produced highly unacceptable results it is concluded that the proposed model provides a convenient and simple means of incorporating hysteretic effects into numerical flow models to provide significant improvement in prediction accuracy with little additional effort and with minimal data requirements

positivity preserving numerical schemes for lubrication type equations lubrication equations are fourth order degenerate diffusion equations of the form h t nabla cdot f h nabla delta h 0 describing thin films or liquid layers driven by surface tension recent studies of singularities in which h to 0 at a point describing rupture of the fluid layer show that such equations exhibit complex dynamics which can be difficult to simulate accurately in particular one must ensure that the numerical approximation of the interface does not show a false premature rupture generic finite difference schemes have the potential to manifest such instabilities especially when underresolved we present new numerical methods in one and two space dimensions that preserve positivity of the solution regardless of the spatial resolution whenever the pde has such a property we also show that the schemes can preserve positivity even when the pde itself is only known to be nonnegativity preserving we prove that positivity preserving finite difference schemes have unique positive solutions at all times we prove stability and convergence of both positivity preserving and generic methods in one and two space dimensions to positive solutions of the pde showing that the generic methods also preserve positivity and have global solutions for sufficiently fine meshes we generalize the positivity preserving property to a finite element framework and show via concrete examples how this leads to the design of other positivity preserving schemes

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

positive harmonic functions vanishing on the boundary for the laplacian in unbounded horn shaped domains denote points x e rd i d 2 by x p 0 z where p o 0 e sd i and z e r let a 0 oo 0 oo be a nondecreasing c2function and define the horn shaped domain q it p 0 z lz 2 by x x z where x e rd and z e r consider a in the horn shaped domain q x x z e rd l izi a ixi where a 0 ox 0 oo is a c2 function since aq is a smooth boundary lipschitz would be enough it follows from a result of hunt and wheeden 3 that every euclidean boundary point on aq may be identified with a distinct martin boundary point thus the interesting part of the martin boundary for a in q is the martin boundary at infinity that is the part of the martin boundary arising from sequences xy c q satisfying limn 0o lxni oo or in terms of positive harmonic functions the part of the martin boundary which corresponds to positive harmonic functions on q which vanish on aq we will give a probabilistic proof of the following theorem the motivation and intuition for which is discussed below received by the editors june 26 1992 1991 mathematics subject classification primary 35j05 secondary 60j50 31c35 31b05 35j15 1994 american mathematical society 0002 9947 94 1 00 25 per page

martin capacity for markov chains the probability that a transient markov chain or a brownian path will ever visit a given set is classically estimated using the capacity of with respect to the green kernel g x y we show that replacing the green kernel by the martin kernel g x y g 0 y yields improved estimates which are exact up to a factor of 2 these estimates are applied to random walks on lattices and also to explain a connection found by lyons between capacity and percolation on trees

geometrical finiteness for hyperbolic groups abstract in this paper we give an account of the notion of geometrical finiteness as applied to discrete groups acting on hyperbolic space of any dimension we prove the equivalence of various definitions of geometrical finiteness and describe the geometry of fundamental domains we give a complete account of when dirichiet domains are finite sided

lebesgue constant minimizing linear rational interpolation of continuous functions over the interval abstract polynomial interpolation between large numbers of arbitrary nodes does notoriously not in general yield useful approximations of continuous functions following 1 we suggest to determine for each set of n 1 nodes another denominator of degree n and then to interpolate every continuous function by a rational function with that same denominator so that the resulting interpolation process remains a linear projection the optimal denominator is chosen so as to minimize the lebesgue constant for the given nodes it has to be computed numerically for that purpose the barycentric representation of rational interpolants which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights is used the optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in 1 we show here how to do so and we present numerical results

on the asymptotic behaviour of first passage times for discussions we investigate the asymptotic behaviour of first passage times of diffusions or birth death processes necessary and sufficient conditions in terms of moments and the speed and scale functions are given for convergence of the first passage times to a normal or an exponential distribution

path integral approach to fluctuations in relativistic transport abstract in this paper we analyze the fluctuations in an approximation to relativistic heat transport theory from a path integral approach we start with a telegraphist type equation for the heat transport derived from a meixner prigogine scheme and construct a classical variational principle for it we use hamiltonian like function to obtain the transition probability between two different non equilibrium states we show that this probability obeys the chapman kolmogorov equation

filtrations and distortion in infinite dimensional algebras abstract a tame filtration of an algebra is defined by the growth of its terms which has to be majorated by an exponential function a particular case is the degree filtration used in the definition of the growth of finitely generated algebras the notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another a geometric analogue is the distortion of the riemannian metric of a lie subgroup when compared to the metric induced from the ambient lie group the distortion of a subalgebra in an algebra also reflects the degree of complexity of the membership problem for the elements of this algebra in this subalgebra one of our goals here is to investigate mostly in the case of associative or lie algebras if a tame filtration of an algebra can be induced from the degree filtration of a larger algebra

numerical investigations of two phase flow with dynamic capillary pressure in porous media via a moving mesh method abstract motivated by observations of saturation overshoot this paper investigates numerical modeling of two phase flow in porous media incorporating dynamic capillary pressure the effects of the dynamic capillary coefficient the infiltrating flux rate and the initial and boundary values are systematically studied using a traveling wave ansatz and efficient numerical methods the traveling wave solutions may exhibit monotonic non monotonic or plateau shaped behavior special attention is paid to the non monotonic profiles the traveling wave results are confirmed by numerically solving the partial differential equation using an accurate adaptive moving mesh solver comparisons between the computed solutions using the brooks corey model and the laboratory measurements of saturation overshoot verify the effectiveness of our approach

fingered flow in two dimensions 2 predicting finger moisture profile an exact integral solution for the moisture profile in growing fingers in sandy soil is derived from richards equation the solution provides moisture content along a finger as a function of position and time and provides applicable results including the calculation of the asymptotic mattic potential of a growing finger and a method of obtaining the unsaturated conductivity in a single experiment the solution is verified experimentally through comparison with measurements of matric potential and moisture content using high speed tensiometers

accurate efficient and iso geometrically flexible collocation methods for phase field models we propose new collocation methods for phase field models our algorithms are based on isogeometric analysis a new technology that makes use of functions from computational geometry such as for example non uniform rational b splines nurbs nurbs exhibit excellent approximability and controllable global smoothness and can represent exactly most geometries encapsulated in computer aided design cad models these attributes permitted us to derive accurate efficient and geometrically flexible collocation methods for phase field models the performance of our method is demonstrated by several numerical examples of phase separation modeled by the cahn hilliard equation we feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo spectral collocation methods and is a viable alternative to classical collocation methods

ergodic properties of boundary actions and the nielsen schreier theory we study the basic ergodic properties ergodicity and conservativity of the action of a subgroup h of a free group f on the boundary pt f with respect to the uniform measure our approach is geometrical and combinatorial and it is based on choosing a system of nielsen schreier generators in h associated with a geodesic spanning tree in the schreier graph x h bs f we give several mod 0 equivalent descriptions of the hopf decomposition of the boundary into the conservative and the dissipative parts further we relate conservativity and dissipativity of the action with the growth of the schreier graph x and of the subgroup h equiv cogrowth of x respectively on the other hand our approach sheds a new light on entirely algebraic properties of subgroups of a free group we also construct numerous examples illustrating the connections between various relevant notions

explicit and almost explicit spectral calculations for diffusion operators abstract the diffusion operator h d 1 2 d d x a d d x b d d x 1 2 exp 2 b d d x a exp 2 b d d x where b x 0 x b a y d y defined either on r 0 with the dirichlet boundary condition at x 0 or on r can be realized as a self adjoint operator with respect to the density exp 2 q x d x the operator is unitarily equivalent to the schrodinger type operator h s 1 2 d d x a d d x v b a where v b a 1 2 b 2 a b we obtain an explicit criterion for the existence of a compact resolvent and explicit formulas up to the multiplicative constant 4 for the infimum of the spectrum and for the infimum of the essential spectrum for these operators we give some applications which show in particular how inf h d scales when a a 0 and b b 0 where and are parameters and a 0 and b 0 are chosen from certain classes of functions we also give applications to self adjoint multi dimensional diffusion operators

comment on on the continuum scale modeling of gravity driven fingers in unsaturated porous media the inadequacy of the richards equation with standard monotonic constitutive relations and hysteretic equations of state authors reply citation deinert m j y parlange k b cady t s steenhuis and j s selker comment on on the continuum scale modeling of gravity driven fingers in unsaturated porous media the inadequacy of the richards equation with standard monotonic constitutive relations and hysteretic equations of state by mehdi eliassi and robert j glass water resour res 39 9 1263 doi 10 1029 2002wr001785 2003

a new approach to the martin boundary via diffusions conditioned to hit a compact set let l generate a transient diffusion x t on rd and let d be an exterior domain let h be the smallest positive solution of lh 0 in d and h 1 on dd define xh t to be the process x t conditioned to hit dd by doob s h transform theory xh t is also a markov diffusion and its generator lh is defined by lhf l h l hf letting rd be the hitting time of dd define the harmonic measure for xh t on dd starting from x e d by i4x dy p r xh trd e dy let xn n 1 c d be a sequence satisfying limp xn i for which i4h converges weakly call two such sequences xn 1 and x 1 equivalent if lim k limn o x we call the set of equivalence classes thus generated the harmonic measure boundary at infinity for lh this boundary is independent of the particular exterior domain d we prove that the harmonic measure boundary at infinity for lh coincides with the martin boundary for l on rd the formal adjoint of the operator l on rd in the case that l generates a reversible diffusion the martin boundaries of l and l coincide and hence the harmonic measure boundary of lh coincides with the martin boundary for l on rd a similar probabilistic description of the martin boundary for l on rd can be given in the nonreversible case these results are then used to give explicit representations of the martin boundaries of l and l for several classes of diffusion processes

on the continuum scale modeling of gravity driven fingers in unsaturated porous media the inadequacy of the richards equation with standard monotonic constitutive relations and hysteretic equations of state we consider the ability of the continuum scale richards equation re with standard monotonic constitutive relations and hysteretic equations of state to model gravity driven fingers in unsaturated porous media and in particular the underlying nonmonotonicity observed along the vertical extent of fingers as published in a number of recent papers finger like solutions with nonmonotonic profiles can be simulated numerically using a downwind averaging method however we find these fingers to be artifacts generated by the combined effects of a truncation error induced oscillation or oversaturation at the wetting front and capillary hysteresis as the oscillation is removed either through grid refinement or the use of inherently monotone schemes the numerically generated fingers disappear and the solution becomes monotonic thus the re along with standard monotonic hydraulic properties does not contain the critical physics required to model gravity driven fingers and must be considered inadequate for unsaturated flow in initially dry highly nonlinear and hysteretic media where these fingers occur

adaptive rational spectral methods for the linear stability analysis of nonlinear fourth order problems this paper presents the application of adaptive rational spectral methods to the linear stability analysis of nonlinear fourth order problems our model equation is a phase field model of infiltration but the proposed discretization can be directly extended to similar equations arising in thin film flows the sharpness and structure of the wetting front preclude the use of the standard chebyshev pseudo spectral method due to its slow convergence in problems where the solution has steep internal layers we discuss the effectiveness and conditioning of the proposed discretization and show that it allows the computation of accurate traveling waves and eigenvalues for small values of the initial water saturation film precursor several orders of magnitude smaller than the values considered previously in analogous stability analyses of thin film flows using just a few hundred grid points

classical field theory and stochastic properties of hyperbolic equations of dissipative processes abstract the set of damped hyperbolic transport equations is one of the wide class of equations for the description of dissipative physical processes deeper understanding into the structure of these physical phenomena can be obtained with the help of the hamiltonian formalism in the present paper we show that the hamilton lagrange formalism can be constructed for these kinds of transport equations we obtain the hamiltonian the canonically conjugate quantities and the poisson bracket expressions for them with this formalism we analyze the statistical properties of path fluctuations in the new conjugated thermodynamic variable space we show that for short times the stochastic behavior under this new scope obeys the chapman kolmogorov relationship

on the porous continuum modeling of gravity driven fingers in unsaturated materials extension of standard theory with a hold back pile up effect 1 the traditional richards equation re in combination with standard monotonic properties constitutive relations and hysteretic equations of state has been shown to lack critical physics required to model gravity driven fingering gdf we extend the re with an experimentally observed hold back pile up hbpu effect not captured in the standard porous continuum re formulation we postulate the hbpu effect is tied to wetting front sharpness and can be mathematically formulated in a variety of ways to include hypodiffusive hyperbolic and mixed spatial temporal forms involving respectively a laplacian a second order derivative in time and a laplacian of a first order derivative in time of the state variables for each we can infer an extended flux relation comprised of the darcy buckingham flux plus an additional component due to the hbpu effect extended flux relations that are mathematically similar to each can be found in the single phase and multiphase flow literature however all with very different underlying conceptualizations of the possible physics

some new procedures for numerical solution of variably saturated flow problems persistent difficulties that arise in forming numerical solutions of variably saturated flow problems include controlling the stability of the nonlinear equation solvers and devising a reliable yet efficient method for determining the positions of seepage surfaces new techniques for addressing these problems are applied to a subdomain finite element discretization of the governing flow equations a series of test problems demonstrates that the techniques are reliable and efficient for a wide variety of problems

invariant σ fields for a class of diffusions the invariant field for a diffusion gives all bounded harmonic functions for the infinitesimal generator of that diffusion we specify the invariant field for a class of two dimensional diffusions and thereby obtain a representation for all bounded harmonic functions for the process when the infinitesimal generator is radially symmetric we obtain the martin boundary this is used to find the invariant field for the corresponding process

harmonic coordinates for diffusions in the plane 1 introduction and main result in this paper we consider coordinate systems for diffusions in the plane which in dimension 1 usually are called natural scales we do not use this term here it is doubtful that in a higher dimensional setting such coordinates will always exist also they serve quite different purposes in dimension 1 these coordinate systems are particularly useful to construct regular diffusion processes in full generality see 21 chapter 5 7 for an account which in higher dimensions is out of reach we have quite a different kind of question in mind imagine that random noise is added to some deterministic dynamical system what is the effect is it negligible or will there be a drastic change in its long term behaviour for such a kind of problem harmonic coordinate systems turn out to be rather useful we shall discuss this in detail in section 2 for the moment let us just explain what we mean by harmonic coordinates some notational conventions vectors are always row vectors with euclidean norm i 1 11 11 is the usual matrix norm is the transpose of the center dot denotes matrix multiplication

a rational spectral collocation method with adaptively transformed chebyshev grid points a spectral collocation method based on rational interpolants and adaptive grid points is presented the rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed chebyshev points the locations of the grid points are adapted to singularities of the underlying solution and the locations of these singularities are approximated by the locations of poles of chebyshev pade approximants numerical experiments on two time dependent problems one with finite time blow up and one with a moving front indicate that the method far outperforms the standard chebyshev spectral collocation method for problems whose solutions have singularities in the complex plane close to 1 1

onsager s regression and the field theory of parabolic transport processes abstract on the basis of a hamiltonian and canonical formalism of simple parabolic transport processes the action for short time intervals can be calculated in the knowledge of action we can examine the stochastic behavior of these processes and we conclude that these are markovian in the space of canonical variables the constructed poisson structure gives us an opportunity to calculate onsager s regression hypothesis

the asymptotic behavior of the solution of the exterior dirichlet problem for brownian motion perturbed by a small parameter drift let l 2a eb v in rd d 2 3 generate a recurrent diffusion for each e 0 where b ei ca rd and let d c rd be an exterior domain then by the recurrence assumption for each f eis c ad there exists a unique solution in the class of bounded solutions to the dirichlet problem leue 0 in d and ue 4i on ad on the other hand by the transience of d dimensional brownian motion there is no uniqueness in the class of bounded solutions for the dirichlet problem 1 au 0 in d and u f on ad since the martin boundary at oo for brownian motion consists of a single point uniqueness is obtained by adding the condition lim1xj u x c we show that u0 x lime o u x exists and satisfies 1 au0 0 in d u0 fi on dd and limb1os uo x c where c is given as follows let pxh denote the measure associated with doob s conditioned brownian motion conditioned to exit d at ad rather than at oo let x inflt 2 0 x t e ad and define the harmonic measure u dy pxh x t e dy then ah lim 4 exists and c jadq y 4 dy we also show that the energy integral jdivui dx when varied over all bounded functions u es w 10o2 d which satisfy u ip on dd takes on its minimum uniquely at u0

on the asymptotic behaviour of solutions of stochastic differential equations in this paper we study the asymptotic behaviour of the solution of the stochastic differential equation dxt g xt dt xt dwt where and g are positive functions and wtis a wiener process we clarify under which conditions xtmay be approximated on xt by means of a deterministic function further the question is treated whether xtconverges in distribution on xt we deal with the ito solution as well as the stratonovitch solution and compare both

the poisson formula for groups with hyperbolic properties the poisson boundary of a group g with a probability measure is the space of ergodic components of the time shift in the path space of the associated random walk via a generalization of the classical poisson formula it gives an integral representation of bounded harmonic functions on g in this paper we develop a new method of identifying the poisson boundary based on entropy estimates for conditional random walks it leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the poisson boundary with natural topological boundaries for several classes of groups word hyperbolic groups and discontinuous groups of isometries of gromov hyperbolic spaces groups with inflnitely many ends cocompact lattices in cartan hadamard manifolds discrete subgroups of semisimple lie groups