solving discrete zero point problems in this paper we present two theorems on the existence of a discrete zero point of a function from the n dimensional integer lattice n to the n dimensional euclidean space n the theorems differ in their boundary conditions for both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps

lie symmetries for integrable evolution equations on the lattice in this work we show how to construct symmetries for the differential difference equations associated with the discrete schrodinger spectral problem we find the whole set of symmetries which in the continuous limit go into the lie point symmetries of the corresponding partial differential equation i e the korteweg de vries equation among these of particular relevance is the non autonomous symmetry which in the continuous limit goes into the dilation symmetry for the corresponding equation unlike the continuous case this symmetry turns out to be a master symmetry thus belonging to the infinite dimensional group of generalized symmetries

solving discrete systems of nonlinear equations abstract we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice z n of the n dimensional euclidean space r n it is assumed that the set is integrally convex which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of z n and each simplex of the triangulation lies in an n dimensional cube of size one with respect to this triangulation we assume that the function satisfies some property that replaces continuity under this property and some boundary condition the function has a zero point to prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations the standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied because the continuity property is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself we apply the main existence result to prove the existence of a pure cournot nash equilibrium in a cournot oligopoly model we further obtain a discrete analogue of the well known borsuk ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem

variable dimension algorithms basic theory interpretations and extensions of some existing methods in this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by van der laan and talman we introduce a new concept primal dual pair of subdivided manifolds and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms our basic model furnishes interpretations to several existing methods lemke s algorithm for the linear complementarity problem its extension to the nonlinear complementarity problem a variable dimension algorithm on conical subdivisions and merrill s algorithm we shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us a method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed

lie symmetries of multidimensional difference equations a method is presented for calculating the lie point symmetries of a scalar difference equation on a two dimensional lattice the symmetry transformations act on the equations and on the lattice they take solutions into solutions and can be used to perform symmetry reduction the method generalizes the one presented in a recent publication for the case of ordinary difference equations in turn it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables

a variable dimension fixed point algorithm and the orientation of simplices a variable dimension algorithm with integer labelling is proposed for solving systems ofn equations inn variables the algorithm is an integer labelling version of the 2 ray algorithm proposed by the author the orientation of lower dimensional simplices is studied and is shown to be preserved along a sequence of adjacent simplices

lie symmetries of soliton equations with self consistent sources via source generation procedure abstract in this paper the source generation procedure sgp proposed by hu and wang x b hu h y wang construction of dkp and bkp equation with self consistent sources inverse problems 22 2006 1903 1920 is utilized to derive lie symmetries of bilinear soliton equations with self consistent sources sescs such as kpescs bkpescs and differential difference kpescs furthermore it is shown that these lie symmetries constitute generators of the corresponding lie symmetry algebras

pfaffianization of the two dimensional toda lattice pfaffianization procedure due to hirota and ohta is applied to the two dimensional toda lattice as a result a pfaffianized version of the two dimensional toda lattice is found

hierarchical parcel swapping representation of turbulent mixing part 1 formulation and scaling properties an economical representation of effects of turbulence on the time evolving structure of diffusive scalar fields is obtained by introducing a hierarchical tree network connecting fluid parcels with effects of turbulent advection represented by swapping pairs of sub trees at rates determined by turbulence time scales associated with the sub trees the fluid parcels reside at the base of the tree the tree structure partitions the fluid parcels into adjacent pairs or more generally p tuples adjacent parcels intermix at rates governed by diffusion time scales based on molecular diffusivities and parcel sizes this simple procedure efficiently accomplishes long standing objectives of turbulent mixing model development such as generating physically based time histories of fluid parcel nearest neighbor encounters and the associated spatial structure of turbulent scalar fields correspondences between features of the hierarchical formulation and turbulent mixing phenomenology both generic and case specific are noted

evaluation of the coulomb force via the fredholm integral equation the electrostatic force between two conducting spheres is solved exactly from coulomb s law the force is evaluated in two steps i the charge distribution on each of the spheres is calculated as a function of separation distance by way of a solution to the fredholm integral equation ii the effect of the known charge distribution is integrated to obtain the electrostatic force evaluation of the electrostatic force is fast because the series expression for the charge distribution is super converging a characteristic trait for some fredholm type equations

computing an integer point of a simplex with an arbitrary starting homotopy like simplicial algorithm abstract an arbitrary starting homotopy like simplicial algorithm is developed for computing an integer point of an n dimensional simplex the algorithm is derived from the use of an integer labeling rule and a triangulation of r n 0 1 and consists of two interchanging phases one phase of the algorithm constitutes a homotopy simplicial algorithm which generates n 1 dimensional simplices in r n 0 1 and the other phase of the algorithm constitutes a pivoting procedure which generates n dimensional simplices in either r n 0 or r n 1 the algorithm varies from one phase to the other when the matrix defining the simplex is in the so called canonical form starting at an arbitrary integer point in r n 0 the algorithm within a finite number of iterations either yields an integer point of the simplex or proves that no such point exists

difference schemes with point symmetries and their numerical tests symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented they have the same three or four dimensional symmetry groups as the original differential equations the new difference schemes are tested as numerical methods the obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost for an example involving a solution with a singularity in the integration region the symmetry preserving scheme contrary to standard ones provides solutions valid beyond the singular point

lie algebra contractions and symmetries of the toda hierarchy the lie algebra l of generalized and point symmetries of the equations in the toda hierarchy is shown to be a semidirect sum of two infinite dimensional lie algebras one perfect the other abelian in the continuous limit the structure of the lie algebra changes a contraction occurs with the lattice spacing as the contraction parameter in particular for the toda equation itself a set of five elements involving both point symmetries and generalized ones contracts to the point symmetry algebra of the potential kdv equation

the existence of lie symmetries for first order analytic discrete dynamical systems if a first order discrete dynamical system dds possesses a continuous symmetry the system is known to be linearizable in this paper we show that a first order analytic dds always possesses a symmetry in a neighbourhood of a hyperbolic fixed point

symmetries of discrete dynamical systems involving two species the lie point symmetries of a coupled system of two nonlinear differential difference equations are investigated it is shown that in special cases the symmetry group can be infinite dimensional in other cases up to ten dimensional the equations can describe the interaction of two long molecular chains each involving one type of atoms

symmetry group of partial differential equations and of differential difference equations the toda lattice versus the korteweg de vries equation the authors correlate the symmetry group of the continuous transformations of the toda lattice to that of the korteweg de vries equation they show how by taking into account the continuous limit of the toda lattice the four parameter symmetry group of the toda lattice is contained in that of the kdv equation by an inverse process discretization of the symmetry group of the kdv equation they find a discrete element of the symmetry group of the toda lattice which gives by symmetry reduction its soliton solution

a vector labeling method for solving discrete zero point and complementarity problems in this paper we establish the existence of a discrete zero point of a function from the n dimensional integer lattice mathbb z n to the n dimensional euclidean space mathbb r n under very general conditions with respect to the behavior of the function the proof is constructive and uses a combinatorial argument based on a simplicial algorithm with vector labeling and lexicographic linear programming pivot steps the algorithm provides an efficient method to find an exact solution we also discuss how to adapt the algorithm for two related problems namely to find a discrete zero point of a function under a general antipodal condition and to find a solution to a discrete nonlinear complementarity problem in both cases the modified algorithm provides a constructive existence proof too we further show that the algorithm for the discrete nonlinear complementarity problem generalizes the well known lemke s method to nonlinear environments an economic application is also presented

a discrete multivariate mean value theorem with applications we establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions a constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling moreover we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions

on the computation of fixed points in the product space of unit simplices and an application to noncooperative n person games in this paper an algorithm based on the principle of simplicial approximation is introduced to compute fixed points of upper semicontinuous point to set mappings from the product space s of unit simplices into itself the algorithm is a modification of an algorithm introduced in an earlier paper the main feature is that it starts with an arbitrary chosen point in s and that the triangulation of s depends on the starting point moreover the algorithm can terminate with a non full dimensional subsimplex yielding a good approximation an application is given for non cooperative n person games where s is the strategy space some computational experiences are given

a unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm we present a unified description of a class of restart fixed point algorithms including merrill s method and several variable dimension algorithms for their implementation on computers based on this description we show that some pivot saving techniques originally developed for the homotopy methods can be applied to the class we also propose a new variable dimension algorithm having 3 n 1 rays along which we can move toward a solution some numerical comparisons of the simplicial restart algorithms merrill s method the 2n method the octahedral method and the new one support that the latter two methods are more efficient than the others

symmetries of the discrete burgers equation a discrete cole hopf transformation is used to derive a discrete burgers equation that is linearizable to a discrete heat equation a five dimensional symmetry algebra is obtained that reduces to the lie point symmetry algebra of the usual burgers equation in the continuous limit this lie algebra is used to obtain explicit invariant solutions

symmetries and first integrals of ordinary difference equations this paper describes a new symmetry based approach to solving a given ordinary difference equation by studying the local structure of the set of solutions we derive a systematic method for determining one parameter lie groups of symmetries in closed form such groups can be used to achieve successive reductions of order if there are enough symmetries the difference equation can be completely solved several examples are used to illustrate the technique for transitive and intransitive symmetry groups it is also shown that every linear second order ordinary difference equation has a lie algebra of symmetry generators that is isomorphic to sl 3 the paper concludes with a systematic method for constructing first integrals directly which can be used even if no symmetries are known

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

umbral calculus difference equations and the discrete schrodinger equation in this paper we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries the method is then applied to the schrodinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space time in this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case moreover systems that are integrable superintegrable or exactly solvable preserve these properties in the discrete case

the korteweg de vries equation and its symmetry preserving discretization the korteweg de vries equation is one of the most important nonlinear evolution equations in the mathematical sciences in this article invariant discretization schemes are constructed for this equation both in the lagrangian and in the eulerian form we also propose invariant schemes that preserve the momentum numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes we find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving galilean transformations which is demonstrated numerically as well

relation between backlund transformations and higher continuous symmetries of the toda equation in this paper we study one aspect of the continuous symmetries of the toda equation namely we establish a correspondence between backlund transformations and continuous symmetries of the toda equation a symmetry transformation acting on a solution of the toda equation can be seen as a superposition of backlund transformations conversely a backlund transformation can be written at least formally as a composition of infinitely many higher symmetry transformations this result reinforces the opinion that the presence of sufficiently many continuous symmetries for discrete equations is an indication of their integrability

efficiency and implementation of simplicial zero point algorithms in this paper we compare the efficiency of several simplicial variable dimension algorithms to do so we first treat the issues of degeneracy and accelerating we present a device for solving degeneracy furthermore we compare several accelerating techniques the technique of iterated quasi newton steps after each major cycle of the simplicial algorithm is implemented in a computer code which is used to compare the efficiency of the n 1 ray 2n ray 2n ray and 3n 1 ray algorithms except for the n 1 ray algorithm the number of function evaluations does not differ very much between the various algorithms it appeared however that the 2n algorithm needs considerably less computation time

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds ideas of a simplicial variable dimension restart algorithm to approximate zero points onrn developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds the algorithm can be considered as a modification of the2n ray zero point finding algorithm onrn it appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n ray algorithm applied to an equivalent zero point problem this is caused by the fact that the algorithm utilizes the complementarity conditions on the variables

binary tree models of high reynolds number turbulence we consider hierarchical models for turbulence that are simple generalizations of the standard gledzer ohkitani yamada shell models e b gledzer dokl akad nauk sssr 209 5 1973 sov phys dokl 18 216 1973 m yamada and k ohkitani j phys soc jpn 56 4210 1987 the density of degrees of freedom is constant in wave number space looking only at this behavior and at the quadratic invariants in the inviscid unforced limit the models can be thought of as systems living naturally in one spatial dimension but being qualitatively similar to hydrodynamics in two 2d and three dimensions we investigated cascade phenomena and intermittency in the different cases we observed and studied a forward cascade of enstrophy in the 2d case

on the existence and approximation of zeroes in this paper we consider the problem of finding zeroes of a continuous functionf from a convex compact subsetu of n to n in the first part of the paper it is proved thatf has a computable zero iff cn n satisfies the nonparallel condition for any two antipodal points on bdcn i e if for anyx bdcn f x f x 0 holds therefore we describe a simplicial algorithm to approximate such a zero it is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm i e the number of times the algorithm switches from a face on bdcn to the face therefore the index of a terminal simplex is defined which equals the local brouwer degree of the function if is full dimensional in the second part of the paper the algorithm is used to generate possibly several approximate zeroes off two sucessive solutions may have both the same or opposite degrees again depending on the number of reflection steps by extendingf u n to a function g from a cube containingu to n the procedure can be applied to any continuous functionf without having any information about the global and local brouwer degrees a priori

lie symmetry preservation and shock capturing methods it is known that some traditional shock capturing schemes including godunov or roe type schemes generate downstream oscillatory wave patterns that cannot be effectively dampened by the dissipation methods they provide the theme of this paper is to understand the formation and behavior at these upstream downstream patterns from the point view of the symmetry inherent in the pde this paper investigates symmetry breaking and spatial oscillations in finite difference solutions for the inviscid burgers and one dimensional navier stokes shock equations using the lax wendroff tvd and nonoscillatory nonfree parameter dissipative schemes symmetry breaking is considered as a candidate for the inherent cause of the nonphysical oscillations in the vicinity of the shock

the jacobi last multiplier for linear partial difference equations we present a discretization of the jacobi last multiplier with some applications to the computation of solutions of linear partial difference equations

point symmetries of generalized toda field theories ii symmetry reduction the lie symmetries of a large class of generalized toda field theories are studied and used to perform symmetry reduction reductions lead to generalized toda lattices on one hand and to periodic systems on the other boundary conditions are introduced to reduce theories on an infinite lattice to those on semi infinite or finite ones

lie symmetries and linearisation of the qrt mapping abstract we study the map u x 2 1 u x 1 u x 2 u x 1 2 u x 1 u x 3 u x 1 introduced by quispel roberts and thompson qrt we show using lie point symmetries under what conditions the qrt mapping can be linearised requiring that the qrt mapping is invariant under the symmetry vector field x x u x x a x b cu du 2 u where b c and d are constants and x is an arbitrary unit periodic function in x we derive conditions on the unknown functions i in the qrt mapping further for these cases of the qrt mapping we explicitly construct two independent integrals of motion ensuring its integrability we also derive its exact solution

symmetry reduction of ordinary finite difference equations using moving frames the technique of equivariant moving frames is incorporated into the classical symmetry reduction method of ordinary finite difference equations using the recurrence relations for the finite difference invariants computations are performed symbolically without relying on the coordinate expressions of the canonical variables and the difference invariants

turbulence large scale sweeping and the emergence of small scale kolmogorov spectra the dynamics of fully developed hydrodynamic turbulence still is a basically unsolved theoretical problem due to the strong coupling long range nonlinearities in the navier stokes equations the present analysis focuses on the small scale fluctuations in a turbulent boundary layer with one external length scale yo after taking a 2 1 d spatiotemporal spectral transform of the fluctuating vorticity fields care is taken of large scale sweeping which arises as a collective zero mode from the nonlinear flow terms the unswept small scale nonlinearities are then shown to be asymptotically locally isotropic i e for wave numbers k by internal consistency which allows to close the nonlinear hierarchy the navier stokes equations without external forcing are integrated to give the spectral response of the fluctuating small scale velocity fields on the presence of a locally isotropic blob of turbulence while it is being swept around over an arbitrary steady state mean velocity profile using viscous boundary conditions at y 0 averaging the response spectrum over all possible orientational configurations and sweep velocities results in a novel self consistency integral for the 4d energy spectrum function the distribution of turbulence sweep velocities is modeled by means of levy type densities having an algebraic tail with power p 1 the generic case which includes von karman s logarithmic mean velocity profile is found to correspond to 1 p 3 asymptotic analysis of the self consistency integral leads to a differential equation which fixes the scaling exponent of the unswept frequency and admits a nonempty integrable and positive definite airy type frequency spectrum ei k k k with so called normal kolmogorov scaling that is 7 3 and 2 3 anomalous scaling is possible for one special mean profile

a new variable dimension algorithm for the fixed point problem we propose a new variable dimension algorithm for solving a system ofn equations inn variables the algorithm solves a sequence of subproblems consisting of the first several equations the dimension of subproblems does not however increase monotonically a convergence condition is also presented

discretization of partial differential equations preserving their physical symmetries a procedure for obtaining a minimal discretization of a partial differential equation preserving all of its lie point symmetries is presented minimal in this case means that the differential equation is replaced by a partial difference scheme involving n difference equations where n is the number of independent and dependent variables we restrict ourselves to one scalar function of two independent variables as examples invariant discretizations of the heat burgers and korteweg de vries equations are presented some exact solutions of the discrete schemes are obtained

continuous symmetries of lagrangians and exact solutions of discrete equations one of the difficulties encountered when studying physical theories in discrete space time is that of describing the underlying continuous symmetries like lorentz or galilei invariance one of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices in a previous article we have classified ordinary difference schemes invariant under lie groups of point transformations the present article is devoted to an invariant lagrangian formalism for scalar single variable difference schemes the formalism is used to obtain first integrals and explicit exact solutions of the schemes equations invariant under two and three dimensional groups of lagrangian symmetries are considered

connection between the spectrum and the moments of the ground state density in n dimensional space we show that a series of recurrent inequalities derived in n 3 have the same formal expressions in any dimension n 2 they are derived from the multipole sum rules and provide us with upper bounds for the moments of the ground state density depending only on the lowest multipole excitation energy these bounds are transformed into approximate recurrent relations by means of an empirical correction factor the 1 r potential and the harmonic oscillator play a key role in establishing this factor which is exact for these two potentials by construction for a large class of potentials we show that this factor tends to 1 as n in such cases at the large n limit the lowest state for each multipole excitation exhausts the sum rule it thus acquires the characteristics of the one phonon excitation typical of the harmonic oscillator

lie point symmetries of differential difference equations we present an algorithm for determining the lie point symmetries of differential equations on fixed non transforming lattices i e equations involving both continuous and discrete independent variables the symmetries of a specific integrable discretization of the krichever novikov equation the toda lattice and toda field theory are presented as examples of the general method

invariant submanifold flows given a lie group acting on a manifold our aim is to analyze the evolution of differential invariants under invariant submanifold flows the constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex applications to integrable soliton dynamics and to the evolution of differential invariant signatures used in equivalence problems and object recognition and symmetry detection in images are discussed

lie discrete symmetries of lattice equations we extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential difference equations as an example of the application of the methods we construct the discrete symmetries of the discrete painleve i equation and of the toda lattice equation

symmetries of discrete dynamical systems differential difference equations of the form un fn t un 1 un un 1 are classified according to their continuous lie point symmetry groups it is shown that for nonlinear equations the symmetry group can be at most seven dimensional the integrable toda lattice is a member of this class and has a four dimensional symmetry group

symmetry classification of diatomic molecular chains a symmetry classification of possible interactions in a diatomic molecular chain is provided for nonlinear interactions the group of lie point transformations leaving the lattice invariant and taking solutions into solutions is at most five dimensional an example is considered in which subgroups of the symmetry group are used to reduce the dynamical differential difference equations to purely difference ones

point symmetries of generalized toda field theories a class of two dimensional field theories with exponential interactions is introduced the interaction depends on two coupling matrices and is sufficiently general to include all toda field theories existing in the literature lie point symmetries of these theories are found for an infinite semi infinite and finite number of fields special attention is accorded to conformal invariance and its breaking

symmetry preserving discretization of ordinary differential equations large symmetry groups and higher order equations ordinary differential equations odes and ordinary difference systems o ss invariant under the actions of the lie groups and of projective transformations of the independent variables x and dependent variables y are constructed the odes are continuous limits of the o ss or conversely the o ss are invariant discretizations of the odes the invariant o ss are used to calculate numerical solutions of the invariant odes of order up to five the solutions of the invariant numerical schemes are compared to numerical solutions obtained by standard runge kutta methods and to exact solutions when available the invariant method performs at least as well as standard ones and much better in the vicinity of singularities of solutions

lie group formalism for difference equations the methods of lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations several examples are analysed one of them being a nonlinear difference equation for the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit

finite size scaling of two point statistics and the turbulent energy cascade generators within the framework of random multiplicative energy cascade models of fully developed turbulence finite size scaling expressions for two point correlators and cumulants are derived taking into account the observationally unavoidable conversion from an ultrametric to an euclidean two point distance the comparison with two point statistics of the surrogate energy dissipation extracted from various wind tunnel and atmospheric boundary layer records allows an accurate deduction of multiscaling exponents and cumulants even at moderate reynolds numbers for which simple power law fits are not feasible the extracted exponents serve as input for parametric estimates of the probabilistic cascade generator various cascade generators are evaluated

lie symmetries and exact solutions of first order difference schemes we show that any first order ordinary differential equation with a known lie point symmetry group can be discretized into a difference scheme with the same symmetry group in general the lattices are not regular ones but must be adapted to the symmetries considered the invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation

construction of partial difference schemes i the clairaut schwarz young theorem on the lattice difference operators in several variables do not always commute their commutation depends on the lattice equations in this paper we study the conditions over a lattice to allow commutativity of these difference operators and their implications for the discretization of partial differential equations

a class of simplicial restart fixed point algorithms without an extra dimension in an earlier paper we introduced an algorithm for approximating a fixed point of a mapping on the product space of unit simplices ideas of that paper are used to construct a class of triangulations ofrn more precisely for somek 1 k n and positive integersm1 mk with sumn a triangulation ofrn is obtained by triangulating the cells which are formed by taking the product of given triangulations ofrmj j 1 k the triangulation of each cell will be defined in relation to an arbitrarily chosen pointv inrn being the starting point of the algorithm fork n we obtain thek triangulation originally due to todd each element of the class can be used to find a simplex which approximates a fixed point of a mapping onrn by generating a unique path of adjacent simplices of variable dimension starting with the pointv

symmetries and conditional symmetries of differential difference equations two different methods of finding lie point symmetries of differential difference equations are presented and applied to the two dimensional toda lattice continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations in particular to differential equations of the delay type the concept of conditional symmetries is extended from purely differential to differential difference equations and shown to incorporate backlund transformations

a modified integer labeling for complementarity algorithms nonlinear complementary problems can be solved by algorithms based on the principle of simplicial approximation we present such an algorithm that allows the choice of an arbitrary starting point without using an artificial labeling or an extra dimension furthermore the relation between the grid size of the triangulation and the accuracy of an approximate solution is investigated and it is shown that a combination of the algorithm with newton steps yields better accuracy finally the algorithm is adapted for solving systems of equations