transformation formula for a double clausenian hypergeometric series its q analogue and its invariance group a transformation formula for a double basic hypergeometric series of type 1 2 20 2 2 is derived this transformation yields a double series analogue of sears transformation for a terminating 3 2 series in the limit q 1 the formula reduces to a transformation for a terminating double clausenian hypergeometric series of unit argument one of the proper kampe de feriet series f1 2 20 2 2 1 1 this formula is a double series analogue of whipple s terminating 3f2 transformation this transformation gives rise to a transformation group the invariance group acting on the parameters of the double series the invariance group is examined and shown to be a subgroup of a double copy of the symmetries of the square

brief constrained linear state estimation a moving horizon approach this article considers moving horizon strategies for constrained linear state estimation additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states noise and other variables formulating a linear state estimation problem with inequality constraints however prevents recursive solutions such as kalman filtering and consequently the estimation problem grows with time as more measurements become available to bound the problem size we explore moving horizon strategies for constrained linear state estimation in this work we discuss some practical and theoretical properties of moving horizon estimation we derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate we also discuss smoothing strategies for moving horizon estimation our framework is solely deterministic

on a maximum principle for inverse monotone matrices abstract let 1 n be a given vector with positive coordinates a matrix a is said to satisfy the maximum principle mp if ax y y 0 imply x 0 and max 1 i n i i max i n y i i where n y is the set of indices such that y is positive for an invertible matrix a with positive inverse the mp is characterized geometrically by means of the behavior under a 1 of convex boundary parts of the simplex generated in r n by permissible multiples of the unit coordinate vectors some sufficient conditions and applications to m matrices are given

some formulae for double clausenian functions abstract for certain f q s s p r r x y functions with p r 3 and q s 2 reduction and transformation formulae under various conditions upon variables and parameters are derived from an euler integral representation that requires one relation between parameters

transformation and summation formulas for double hypergeometric series abstract a number of new transformation formulas for double hypergeometric series are presented the series appearing here are the so called kampe de feriet functions of type f1 1 20 3 4 1 1 and f0 2 21 2 2 1 1 the transformation formulas relate such double series to a single hypergeometric series of 4f3 1 type by specializing certain parameters a list of new summation formulas for f0 2 21 2 2 1 1 series is obtained the origin of the results comes from studying symmetries of the 9 j coefficient appearing in quantum theory of angular momentum

reduction and transformation formulae for bivariate basic hypergeometric series the main object of this paper is to establish several bivariate basic hypergeometric series identities by means of elementary series manipulation some of them can be applied to yield transformation and reduction formulae for q kampe de feriet functions

transformation and summation formulae for kampe de feriet series abstract euler s two transformations are employed to establish several general transformation formulae for double hypergeometric series these transformations yield further a number of reduction formulae for kampe de feriet series of types f 1 0 1 1 f 1 0 0 2 and f 1 0 2 0 by specializing certain parameters a list of new summation formulae for these kampe de feriet series are obtained

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

a solution for the partitioning problem in partition based moving horizon estimation in addition to being observable systems composed of interconnected subsystems must typically exhibit a particular subsystem structure if their state is to be estimated successfully by a partition based moving horizon estimator pmhe suitable subsystem structures also called system partitionings can be unknown may not be desirable for a particular application or may not exist at all in this contribution we present the first pmhe without any restrictions on the system partitioning we analyze the convergence of its iterations at each time step and the stability of its estimation error dynamics the benefits of our solution are illustrated on a numerical example

multiple hypergeometric functions and 9 j coefficients it is well known that the 9 j recoupling coefficient appearing in the quantum theory of angular momentum has 72 symmetries however the triple sum series expression for the 9 j coefficient exhibits none of these symmetries here a stretched 9 j coefficient for which a closed form single term expression exists is considered and the type of summation theorems the triple sum series reduces is investigated for any of the 72 symmetries apart from well known single summation theorems for hypergeometric functions this analysis gives rise to new summation theorems for double and triple hypergeometric functions

some families of hypergeometric polynomials and associated integral representations the main object of this paper is to investigate several general families of hypergeometric polynomials and their associated single double and triple integral representations some known or new consequences of the general results presented here involving such classical orthogonal polynomials as the jacobi laguerre hermite and bessel polynomials and various other relatively less familiar hypergeometric polynomials are also considered each of the integral representations which are derived in this paper may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials

hypergeometric series related to the 9 j coefficient of su 1 1 abstract we study various expressions for the 9 j coefficient of su 1 1 with an emphasis on multiple hypergeometric series an orthogonal polynomial in two discrete variables is associated to the 9 j coefficient some applications are considered dealing with formulas in which the 9 j coefficient appears as a connection or expansion coefficient and discussing some transformation formulas for double and triple hypergeometric series of unit argument

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

suboptimal receding horizon estimation via noise blocking abstract for discrete time linear systems we propose a suboptimal approach to constrained estimation so that the associated computation burden is reduced this is achieved by enforcing a move blocking mb structure in the estimated process noise sequence pns we show that full information estimation fie and receding horizon estimation rhe with mb are both stable in the sense of an observer the techniques in proving stability are inspired by those that have been proposed for standard rhe to be specific stability results are mainly achieved by i carefully embellishing the general assumptions for standard rhe to accommodate the mb requirement ii exploiting the principle of optimality as well as convexity of the quadratic programs qps associated with fie and rhe iii relying on the fact that the kalman filter is the best linear estimator in the least squares sense a crucial requirement in achieving stability for mb rhe is that the segment structure ss of the pns of mb fie for the optimization steps within the receding horizon i e steps between t n and t 1 has to be enforced in the mb rhe optimization as a result the mb rhe strategy becomes a dynamic estimator with a periodically varying computational complexity the theoretical results have been illustrated with examples

series identities and associated families of generating functions the authors investigate several families of double series identities as well as their known or new consequences involving various hypergeometric functions in one and two variables a number of associated generating function relationships involving certain classes of hypergeometric polynomials are also considered

new families of generating functions for certain class of three variable polynomials abstract recently srivastava ozarslan and kaanoglu have introduced certain families of three and two variable polynomials which include lagrange and lagrange hermite polynomials and obtained families of two sided linear generating functions between these families h m srivastava m a ozarslan c kaanoglu some families of generating functions for a certain class of three variable polynomials integr transform spec funct ifirst 2010 1 12 the main object of this investigation is to obtain new two sided linear generating functions between these families by applying certain hypergeometric transformations furthermore more general families of bilinear bilateral multilateral finite series relationships and generating functions are presented for them

some families of generating functions for the multiple orthogonal polynomials associated with modified bessel k functions the main object of this paper is to derive several substantially more general families of bilinear bilateral and mixed multilateral finite series relationships and generating functions for the multiple orthogonal polynomials associated with the modified bessel k functions also known as macdonald functions some special cases of the above statements are also given

moving horizon estimation for discrete time linear systems with binary sensors algorithms and stability results abstract the paper addresses state estimation for linear discrete time systems with binary threshold measurements a moving horizon estimation mhe approach is followed and different estimators characterized by two different choices of the cost function to be minimized and or by the possible inclusion of constraints are proposed specifically the cost function is either quadratic when only the information pertaining to the threshold crossing instants is exploited or piece wise quadratic when all the available binary measurements are taken into account stability results are provided for the proposed mhe algorithms in the presence of unknown but bounded disturbances and measurement noise performance of the proposed techniques is also assessed by means of simulation examples

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

weakened acute type condition for tetrahedral triangulations and the discrete maximum principle we prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the poisson equation with the dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain this result represents a weakened form of the acute type condition for the three dimensional case

ordinary and basic bivariate hypergeometric transformations associated with the appell and kampe de feriet functions by means of formal power series method and series rearrangement technique the authors investigate and establish several ordinary as well as basic or q bivariate hypergeometric transformation and reduction formulas these latter formulas may be considered as basic or q extensions of some recent results considered by lee et al internat j comput math 78 2001 303 321 and karlsson et al internat j math statist sci 9 2000 211 226 the bivariate hypergeometric series which are considered in this paper are closely related to the familiar appell and kampe de feriet functions

a certain class of incomplete elliptic integrals and associated definite integrals in many seemingly diverse physical contexts including for example certain radiation field problems studies of crystallographic minimal surfaces the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk studies of elliptical crack problems in fracture mechanics and so on a remarkably large number of general families of elliptic type integrals and indeed also many definite integrals of such families with respect to their modulus or complementary modulus are known to arise naturally motivated essentially by these and many other potential avenues of their applications we present here a systematic account of the theory of a certain family of incomplete elliptic integrals in a unified and generalized manner by means of the familiar riemann liouville fractional differintegral operators we obtain several explicit hypergeometric representations and apply these representations with a view to deriving various associated definite integrals not only with respect to the modulus or complementary modulus but also with respect to the amplitude of the incomplete elliptic integrals involved therein

a note on the convergence of certain families of multiple hypergeometric series abstract in this sequel to an earlier work on the subject the authors present a further analysis of the convergence problem of kampe de feriet s double hypergeometric series and also of lauricella s multiple hypergeometric series when the arguments take on values on the boundaries of the known open regions of convergence of these series

finite volume method for coupled subsurface flow problems i darcy problem abstract the article introduces a finite volume method for the darcy problem in heterogeneous anisotropic media the method is based on the mixed formulation for the pressure and its gradient the method is stable despite collocation of both pressure and its gradient at cell centers and demonstrates the first order convergence on numerous benchmarks as well as good monotonicity property the method produces quasi definite matrix which is numerically shown to have good asymptotics of the condition number our flux discretization method is a realization of our more general concept of stable flux discretization for saddle point systems with vector of several unknowns in this paper this vector is composed of pressure and its gradient and the saddle point system is the mixed formulation of the darcy problem

new transformation and reduction formulae for double q clausen series abstract the sears transformations for non terminating 3 2 series are applied to derive several general series transformations for another two double q clausen series of type 2 0 1 2 and 2 0 0 3 further new transformations and reduction formulae are established for the basic clausen hypergeometric series in particular several bibasic reduction formulae are obtained

on maximum principles for m operators abstract for m matrices a condition to satisfy the maximum principle for inverse column entries is known we generalize this result concerning a more general maximum principle for m operators on r n ordered by some cone as well as to a certain extent for m operators on infinite dimensional ordered normed spaces

two hypergeometric summation formulae related to 9 j coefficients two summation formulae for special kampe de feriet functions f1 10 3 with variables 1 1 are proved

monotonic solution of heterogeneous anisotropic diffusion problems anisotropic problems arise in various areas of science and engineering for example groundwater transport and petroleum reservoir simulations the pure diffusive anisotropic time dependent transport problem is solved on a finite number of nodes that are selected inside and on the boundary of the given domain along with possible internal boundaries connecting some of the nodes an unstructured triangular mesh that attains the generalized anisotropic delaunay condition for all the triangle sides is automatically generated by properly connecting all the nodes starting from an arbitrary initial one the control volume of each node is the closed polygon given by the union of the midpoint of each side with the anisotropic circumcentre of each final triangle a structure of the flux across the control volume sides similar to the standard galerkin finite element scheme is derived a special treatment of the flux computation mainly based on edge swaps of the initial mesh triangles is proposed in order to obtain a stiffness m matrix system that guarantees the monotonicity of the solution the proposed scheme is tested using several literature tests and the results are compared with analytical solutions as well as with the results of other algorithms in terms of convergence order computational costs are also investigated

certain classes of generating functions for the jacobi and related hypergeometric polynomials abstract for a certain class of generalized hypergeometric polynomials the authors first derive a general theorem on bilinear bilateral and mixed multilateral generating functions and then apply these generating functions in order to deduce the corresponding results for the classical jacobi and laguerre polynomials they also consider several linear generating functions for these polynomials as well as for some multivariable jacobi and multivariable laguerre polynomials which were investigated in recent years some of the linear generating functions presented in this paper are associated with the stirling numbers of the second kind

stretched 9 j coefficients and summation theorems it is well known that the 9 j coefficient has 72 symmetries and that its simplest known form is the triple sum series which exhibits none of these symmetries a study of the expressions for the stretched 9 j coefficients stretched in the sense that that one or more of its six triangle inequality components are stretched via the triple sum series shows the existence of the 2f1 1 3f2 1 and the 4f3 1 summation theorems and reveals some new summation theorems as well

on extended hurwitz lerch zeta function abstract this paper investigates an extended form of a beta function b p q x y we first study the convergence problem of the function b p q x y and consider the completely monotonic and log convex properties of this function as a result we obtain a pair of laguerre type inequalities next we provide a new double integral representation for the function b p q x y subsequently we consider the convergence problem of the extended hurwitz lerch zeta function z s a p q defined by its series representation upon using the series manipulation techniques we obtain two series identities we also find various integral representations for the function z s a p q lastly we apply fourier analysis to the function z a z s a p q and obtain a lindelof wirtinger type expansion some interesting and promising results are also illustrated

on matrices satisfying a maximum principle with respect to a cone abstract a matrix equation a x y is considered in the space r n that is ordered by a cone k in case k r n it is known that a matrix a is said to satisfy the maximum principle if a is invertible and for each y r n 0 the solution x belongs to r n and is such that xi max1 k nxk and yi 0 for some i this concept is generalized to finitely generated and to circular cones k r n this is achieved by evaluating x and y with the aid of elements of a given base for the polar cone k of k the maximum principle is characterized geometrically by means of the behavior under a 1 of convex boundary parts of a base for k a weighted maximum principle is investigated and an infinite dimensional example is indicated

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

fractional integration of the h function of several variables abstract the main object of the present paper is to derive a number of key formulas for the fractional integration of the multivariable h function which is defined by a multiple contour integral of mellin barnes type each of the general eulerian integral formulas obtained in this paper are shown to yield interesting new results for various families of generalized hypergeometric functions of several variables some of these applications of the key formulas would provide potentially useful generalizations of known results in the theory of fractional calculus

transformations of multipleq series with quasi arbitrary terms abstract the main object of this paper is to present several presumably new transformations of series whose terms involve basic or q pochhammer symbols together with an essentially arbitrary factor each of these general q series transformations is shown to be related rather closely to a certain basic hypergeometric summation or transformation formula relevant connections of many of the multiple q series transformations with various known results are also considered briefly

constrained state estimation for nonlinear discrete time systems stability and moving horizon approximations state estimator design for a nonlinear discrete time system is a challenging problem further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances one strategy for constrained state estimation is to employ online optimization using a moving horizon approximation we propose a general theory for constrained moving horizon estimation sufficient conditions for asymptotic and bounded stability are established we apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation examples are used to illustrate the benefits of constrained state estimation our framework is deterministic

maximum principle in linear finite element approximations of anisotropic diffusion convection reaction problems a mesh condition is developed for linear finite element approximations of anisotropic diffusion convection reaction problems to satisfy a discrete maximum principle loosely speaking the condition requires that the mesh be simplicial and mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix where h h denotes the mesh size and varvec b b and c c are the coefficients of the convection and reaction terms in two dimensions the condition can be replaced by a weaker mesh condition an mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 perturbation of a generalized delaunay condition these results include many existing mesh conditions as special cases numerical results are presented to verify the theoretical findings

transformation formulas for double hypergeometric series related to 9 j coefficients and their basic analogs in a recent paper alisauskas deduced different triple sum expressions for the 9 j coefficient of su 2 and suq 2 for a singly stretched 9 j coefficient these reduce to different double sum series using these distinct series we deduce a set of new transformation formulas for double hypergeometric series of kampe de feriet type and their basic analogs these transformation formulas are valid for rather general parameters of the series although a common feature is that all the series appearing here are terminating it is also shown that the transformation formulas deduced here generate a group of transformation formulas thus yielding an invariance group or symmetry group of particular double series

some multiple hypergeometric transformations and associated reduction formulas the main object of the present paper is to derive various classes of double series identities and to show how these general results would apply to yield some known or new reduction formulas for the appell kampe de feriet and lauricella hypergeometric functions of several variables a number of closely related linear generating functions for the classical jacobi polynomials are also investigated

a constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems nonlinear constrained finite element approximations to anisotropic diffusion problems are considered starting with a standard linear or bilinear galerkin discretization the entries of the stiffness matrix are adjusted so as to enforce sufficient conditions of the discrete maximum principle dmp an algebraic splitting is employed to separate the contributions of negative and positive off diagonal coefficients which are associated with diffusive and antidiffusive numerical fluxes respectively in order to prevent the formation of spurious undershoots and overshoots a symmetric slope limiter is designed for the antidiffusive part the corresponding upper and lower bounds are defined using an estimate of the steepest gradient in terms of the maximum and minimum solution values at surrounding nodes the recovery of nodal gradients is performed by means of a lumped mass l 2 projection the proposed slope limiting strategy preserves the consistency of the underlying discrete problem and the structure of the stiffness matrix symmetry zero row and column sums a positivity preserving defect correction scheme is devised for the nonlinear algebraic system to be solved numerical results and a grid convergence study are presented for a number of anisotropic diffusion problems in two space dimensions

on maximum principles for monotone matrices abstract for monotone matrices and h matrices maximum principles are defined and necessary and sufficient conditions are derived for these principles to hold

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

an anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics petroleum engineering and image processing standard numerical methods can produce spurious oscillations when they are used to solve those problems a common approach to avoid this difficulty is to design a proper numerical scheme and or a proper mesh so that the numerical solution validates the discrete counterpart dmp of the maximum principle satisfied by the continuous solution a well known mesh condition for the dmp satisfaction by the linear finite element solution of isotropic diffusion problems is the non obtuse angle condition that requires the dihedral angles of mesh elements to be non obtuse in this paper a generalization of the condition the so called anisotropic non obtuse angle condition is developed for the finite element solution of heterogeneous anisotropic diffusion problems the new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem several variants of the new condition are obtained based on one of them two metric tensors for use in anisotropic mesh generation are developed to account for dmp satisfaction and the combination of dmp satisfaction and mesh adaptivity numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors

an iterative partition based moving horizon estimator with coupled inequality constraints we propose an iterative partition based moving horizon state estimator for large scale linear systems that consist of interacting subsystems every subsystem estimates its own state and disturbance variables taking into account the estimates received from neighboring subsystems compared to other partition based moving horizon estimators the proposed method has two unique features it can handle coupled inequality constraints on the estimated variables and its state estimates come arbitrarily close to the optimal state estimates of a centralized moving horizon estimator the applicability and performance of the proposed method are demonstrated on a numerical example and convergence and asymptotic stability are rigorously proven

brief paper moving horizon partition based state estimation of large scale systems this paper presents three novel moving horizon estimation mhe methods for discrete time partitioned linear systems i e systems decomposed into coupled subsystems with non overlapping states the mhe approach is used due to its capability of exploiting physical constraints on states and noise in the estimation process in the proposed algorithms each subsystem solves reduced order mhe problems to estimate its own state and different estimators have different computational complexity accuracy and transmission requirements among subsystems in all cases proper tuning of the design parameters i e the penalties on the states at the beginning of the estimation horizon guarantees convergence of the estimation error to zero numerical simulations demonstrate the viability of the approach

global and local refinement techniques yielding nonobtuse tetrahedral partitions preservation of basic qualitative properties for example the maximum principle ofthe solution of partial differential equations by its finite element approximations is an important goal in mathematical modelling and simulation nonobtuse tetrahedral partitions and linear finite elements guarantee the validity of the discrete analogues of the maximum principle for a wide class of parabolic and elliptic problems in order to get more accurate approximation we often need to refine the used partitions globally or locally in this paper we first propose two variants of global refinement techniques which produce nonobtuse face to face tetrahedral partitions second we present a new local refinement technique which generates nonobtuse face to face tetrahedral partitions in a neighbourhood of a given vertex

the cutoff method for the numerical computation of nonnegative solutions of parabolic pdes with application to anisotropic diffusion and lubrication type equations the cutoff method which cuts off the values of a function less than a given number is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations a convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations two applications are investigated linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication type equations for which it is unclear if the convergence analysis applies the numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature the convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions cutoff can also be used with other discretization methods such as collocation finite volume finite element and spectral methods and for the computation of positive solutions