frozen percolation in two dimensions aldous math proc camb philos soc 128 465 477 2000 introduced a modification of the bond percolation process on the binary tree where clusters stop growing freeze as soon as they become infinite we investigate the site version of this process on the triangular lattice where clusters freeze as soon as they reach l infty diameter at least n for some parameter n we show informally speaking that in the limit n rightarrow infty the clusters only freeze in the critical window of site percolation on the triangular lattice hence the fraction of vertices that eventually i e at time 1 are in a frozen cluster tends to 0 as n goes to infinity we also show that the diameter of the open cluster at time 1 of a given vertex is with high probability smaller than n but of order n this shows that the process on the triangular lattice has a behaviour quite different from aldous process we also indicate which modifications have to be made to adapt the proofs to the case of the n parameter frozen bond percolation process on the square lattice this extends our results to the square lattice and answers the questions posed by van den berg et al random struct algorithms 40 220 226 2012

stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations abstract this paper deals with the stability analysis of one step methods for the numerical solution of initial value problems both stiff ordinary and partial differential equations are included the problem is considered how to restrict the stepsize in the methods in order that they behave stable we review and extend some recent results on this problem that are based on the use of stability regions in the complex plane we focus on differential equations that are essentially more general than the classical test equation u t u t with a complex constant further the emphasis is on stability of the methods with respect to the maximum norm

on the noetherianity of some associative finitely presented algebras abstract we consider finitely generated associative algebras over a fixed field k of arbitrary characteristic for such an algebra a we impose some structural restrictions we call a strictly ordered we are interested in the implication of strict order on a for its noetherian properties in particular we prove that if a is a graded standard finitely presented strictly ordered algebra then a is left noetherian if and only if it is almost commutative in this case a has polynomial growth

zero stability properties of the three ordinate variable stepsize variable formula methods consider the systemy f x y y a x a b y r n wheref is continuous and lipschitzian with respect to the second argument very often linear multistep variable stepsize variable formula methods lm vsvfm s are used to computey k y xk on the points of the grid a x 0 x1 x2 xn b the general lm vsvfm is based on formulae of the following type y k sum limits i 1 s k alpha i bar h k sk y k i sum limits i 0 s k h k i beta i bar h k s k f x k i y k i whereh k xk xk 1 bar h k h k h k 1 ldots h k s k s k k k 1 1 n the coefficients i and i depend on the lasts k 1 stepsizes and on the formula used at stepk only the zero stability properties of some special classes of lm vsvfm s as for example those based on adams formulae were investigated in the literature a class of three ordinate lm vsvfm s is defined in this paper some results concerning the zero stability properties of these methods are proved it is shown that some well known results are simple corollaries of the results found for the three ordinate lm vsvfm s it is easily seen that similar results hold for the corresponding one leg vsvfm s finally the use of the theoretical results in the practical implementations is briefly discussed

numerical integrators for the hybrid monte carlo method we construct numerical integrators for hamiltonian problems that may advantageously replace the standard verlet time stepper within hybrid monte carlo and related simulations past attempts have often aimed at boosting the order of accuracy of the integrator and or reducing the size of its error constants order and error constant are relevant concepts in the limit of vanishing step length we propose an alternative methodology based on the performance of the integrator when sampling from gaussian distributions with not necessarily small step lengths we construct new splitting formulae that require two three or four force evaluations per time step limited proof of concept numerical experiments suggest that the new integrators may provide an improvement on the efficiency of the standard verlet method especially in problems with high dimensionality

high order exponential operator splitting methods for time dependent schrodinger equations in this paper we deduce high order error bounds for exponential operator splitting methods the employed techniques are specific to linear differential equations of the form u t a u t b u t t geq 0 involving an unbounded operator a in particular evolutionary schrodinger equations with sufficiently regular initial values are included in the analysis

outlets of 2d invasion percolation and multiple armed incipient infinite clusters we study invasion percolation in two dimensions focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters iic s first we compute the exact decay rate of the distribution of both the weight of the k th outlet and the volume of the k th pond next we prove bounds for all moments of the distribution of the number of outlets in an annulus this result leads to almost sure bounds for the number of outlets in a box b 2 n and for the decay rate of the weight of the k th outlet to pc we then prove existence of multiplearmed iic measures for any number of arms and for any color sequence which is alternating or monochromatic we use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin

high order difference schemes with reduced dispersion for hyperbolic differential equations abstract we investigate difference schemes for systems of first order hyperbolic differential equations in two space dimensions possessing the following characteristics 1 the spatial discretizations are fourth order accurate 2 the time discretization is of explicit runge kutta type and is also fourth order accurate 3 the scaled stability boundary is approximately 1 2 2 4 the weights in the space discretizations and the runge kutta parameters can be adapted in order to reduce the dispersion of dominant fourier components this method is illustrated by applying it to the shallow water equations simulating the motion of water in a shallow sea due to tidal forces since in such problems the dominant frequencies in the solution are known in advance the method can take fully advantage of the possibility to tune the various parameters to these dominant frequencies

explicit nordsieck methods with extended stability regions abstract we describe the construction of explicit nordsieck methods of order p and stage order q p with large regions of absolute stability we also discuss error propagation and estimation of local discretization errors the error estimators are derived for examples of general linear methods constructed in this paper some numerical experiments are presented which illustrate the effectiveness of proposed methods

infinite clusters in percolation models the qualitative nature of infinite clusters in percolation models is investigated the results which apply to both independent and correlated percolation in any dimension concern the number and density of infinite clusters the size of their external surface the value of their total surface to volume ratio and the fluctuations in their density in particular it is shown thatn0 the number of distinct infinite clusters is either 0 1 or and the casen0 which might occur in sufficiently high dimension is analyzed

imex extensions of linear multistep methods with general monotonicity and boundedness properties for solving hyperbolic systems with stiff sources or relaxation terms time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms in this paper we consider implicit explicit imex multistep methods suitable methods will be constructed based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations numerical comparisons are made with several implicit explicit runge kutta methods

invasion percolation and the incipient infinite cluster in 2d we establish two links between two dimensional invasion percolation and kesten s incipient infinite cluster iic we first prove that the kth moment of the number of invaded sites within the box n n n n is of order n2 n k for k 1 where n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n this improves a result of y zhang we show that the size of the invaded region when scaled by n2 n is tight

high order imex spectral schemes for computing the dynamics of systems of nonlinear schrodinger gross pitaevskii equations the aim of this paper is to build and validate some explicit high order schemes both in space and time for simulating the dynamics of systems of nonlinear schrodinger gross pitaevskii equations the method is based on the combination of high order implicit explicit imex schemes in time and fourier pseudo spectral approximations in space the resulting imexsp schemes are highly accurate efficient and easy to implement they are also robust when used in conjunction with an adaptive time stepping strategy and appear as an interesting alternative to time splitting pseudo spectral tssp schemes finally a complete numerical study is developed to investigate the properties of the imexsp schemes in comparison with tssp schemes for one and two components systems of gross pitaevskii equations

letter to the editor a simple estimation for the spectral radius of block h matrices simple bounds for the spectral radius of an h matrix and a block h matrix are presented respectively i e a 2maxi aii for an h matrix a aij and a maxi aii aii 1 for a block h matrix a aij where a denote the spectral radius

convergence analysis of high order time splitting pseudospectral methods for nonlinear schrodinger equations in this work the issue of favorable numerical methods for the space and time discretization of low dimensional nonlinear schrodinger equations is addressed the objective is to provide a stability and error analysis of high accuracy discretizations that rely on spectral and splitting methods as a model problem the time dependent gross pitaevskii equation arising in the description of bose einstein condensates is considered for the space discretization pseudospectral methods collocated at the associated quadrature nodes are analyzed for the time integration high order exponential operator splitting methods are studied where the decomposition of the function defining the partial differential equation is chosen in accordance with the underlying spectral method the convergence analysis relies on a general framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part essential tools in the derivation of a temporal global error estimate are furthe

stability and accuracy of time discretizations for initial value problems this paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems here the treatment is extended to cover also implicit schemes and by placing the accuracy of the schemes into a more central position in the discussion general method free statements are again obtained more specialized results are additionally given for linear multistep methods and for the taylor series method

mathematical model for studying the sulphur pollution over europe the long range transport of sulphur pollutants in the atmosphere is an important environmental problem a mathematical model consisting of a system of partial differential equations has been developed at the danish agency of environmental protection for studying this phenomenon the model is handled numerically a pseudospectral fourier algorithm is used in the space discretization phase a variable stepsize variable formula method vsvfm based on predictor corrector schemes may be specified during the time discretization part the time integration formulae are especially constructed so that a the predictor corrector schemes have good absolute stability properties on the imaginary axis and b the fundamental properties consistency zero stability and convergence of the global vsvfm are not affected by the changes of the stepsize and or the predictor corrector scheme both the accuracy requirements and the stability requirements are taken into account in the stepsize selection strategy as well as in the formula selection strategy this double check leads to a great efficiency in the numerical treatment of problems arising in atmospheric environments this is demonstrated by many numerical results the question of the reliability of the results obtained in runs of real physical problems is discussed some results obtained in the comparison of calculated results with measurements are reported an illustration where the model is applied in studying the air pollution in the area around denmark due to danish sources is presented

quadratic algebras of skew type and the underlying monoids abstract we consider algebras over a field k defined by a presentation k x1 xn r where r consists of n 2 square free relations of the form xixj xkxl with every monomial xixj i j appearing in one of the relations certain sufficient conditions for the algebra to be noetherian and pi are determined for this we prove more generally that right noetherian algebras of finite gelfand kirillov dimension defined by homogeneous semigroup relations satisfy a polynomial identity the structure of the underlying monoid defined by the same presentation is described this is used to derive information on the prime radical and minimal prime ideals some examples are described in detail earlier gateva ivanova and van den bergh and jespers and okninski considered special classes of such algebras in the contexts of noetherian algebras grobner bases finitely generated solvable groups semigroup algebras and set theoretic solutions of the yang baxter equation

high order linear multistep methods with general monotonicity and boundedness properties we consider linear multistep methods that possess general monotonicity and boundedness properties strict monotonicity in terms of arbitrary starting values for the multistep schemes is only valid for a small class of methods under very stringent step size restrictions this makes them uncompetitive with the strong stability preserving ssp runge kutta methods by relaxing these strict monotonicity requirements a larger class of methods can be considered including many methods of practical interest in this paper we construct linear multistep methods of high order up to six that possess relaxed monotonicity or boundedness properties with optimal step size conditions numerical experiments show that the new schemes perform much better than the classical monotonicity preserving multistep schemes moreover there is a substantial gain in efficiency compared to recently constructed ssp runge kutta ssprk methods

local time on the exceptional set of dynamical percolation and the incipient infinite cluster in dynamical critical site percolation on the triangular lattice or bond percolation on z2 we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster we show that at a typical time with respect to this measure the percolation configuration has the law of kesten s incipient infinite cluster in the most technical result of this paper we show that on the other hand at the first exceptional time the law of the configuration is different we believe that the two laws are mutually singular but do not show this we also study the collapse of the infinite cluster near typical exceptional times and establish a relation between static and dynamic exponents analogous to kesten s near critical relation

relations between invasion percolation and critical percolation in two dimensions we study invasion percolation in two dimensions we compare connectivity properties of the origin s invaded region to those of a the critical percolation cluster of the origin and b the incipient infinite cluster to exhibit similarities we show that for any k 1 the k point function of the first socalled pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin more prominent though are the differences we show that there are infinitely many ponds that contain many large disjoint pc open clusters further for k 1 we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin we finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular

a comparison of different propagation schemes for the time dependent schro dinger equation abstract a comparison of three widely used time propagation algorithms for the time dependent schrodinger equation is described a typical evolution problem is chosen to demonstrate the efficiency and accuracy of the various methods on a numerical grid using a pseudo spectral fft spatial representation for scattering and bound state evolution the methods used second order differencing split operator propagation chebyshev polynomial expansion are discussed in terms of their applicability to various classes of dynamic problems a new method is introduced which is based upon a low order lanczos technique this method appears to offer an accurate and flexible alternative to the existing techniques overall the chebyshev method is recommended for time independent potentials and the lanczos method for time dependent potentials

dimensional reductions for the computation of time dependent quantum expectations we consider dimension reduction techniques for the liouville von neumann equation for the evaluation of the expectation values in a mixed quantum system we describe several existing methods that have appeared in the literature showing the failure of them when the system is scaled up we introduce a new method termed direct expectation values via chebyshev based on evaluation of a trace formula combined with a direct expansion in modified chebyshev polynomials this reduction is highly efficient and does not destroy any information we demonstrate the practical application of the scheme for a nuclear spin system and compare with popular alternatives in nuclear spin dynamics the main goal for simulations is being able to simulate a system with as many spins as possible for this reason it is very important to have an efficient method that scales the least with respect to the number of particles this method may be applied to autonomous quantum problems where the desired outcome of quantum simulation rather than being a full description of the system is only the expectation value of some observables

a fourier method solution for the time dependent schrodinger equation as a tool in molecular dynamics a new method is presented for the solution of the time dependent schrbdinger equation in its application to physical and chemical molecular phenomena the method is based on discretizing space and time on a grid and using the fourier method to produce both spatial derivatives and second order differencing for time derivatives the method conserves norm and energy and preserves quantum mechanical commutation relations one and twodimensional examples where a comparison to analytic results is possible are investigated many phenomena of physics and chemistry have a common dynamical evolution pattern this pattern begins with an initial state which under the influence of the potentials evolves through time to produce a final asymptotic state among such phenomena are chemical reactions photodissociation unimolecular breakdown surface scattering and desorption the present study presents a numerical solution based on quantum mechanics for the common pattern or for the time dependent evolution of the state of the system the importance of exact numerical solutions is twofold first these solutions give a quantitative description of physical problems and also are of use as a bench mark to check approximation methods and to define their range of validity

invasion percolation a new form of percolation theory a new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system the model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance the name invasion percolation is proposed for this new process similarities to and differences from ordinary percolation theory are discussed

increasing the real stability boundary of explicit methods abstract based on the simplest well known integration rules such as the forward euler scheme and the classical runge kutta method an extension is proposed to enlarge the real stability boundary the main characteristic of the resulting schemes is that the computational complexity is hardly increased

noetherian properties of skew polynomial rings with binomial relations in this work we study standard finitely presented associative algebras over a fixed field k a restricted class of skew polynomial rings with quadratic relations considered in an earlier work of m artin and w schelter will be studied we call them binomial skew polynomial algebras we establish necessary and sufficient conditions for such an algebra to be a noetherian domain

an introduction to commutative and noncommutative gro bner bases abstract in 1965 buchberger introduced the notion of grobner bases for a polynomial ideal and an algorithm buchberger algorithm for their computation since the end of the seventies grobner bases have been an essential tool in the development of computational techniques for the symbolic solution of polynomial systems of equations and in the development of effective methods in algebraic geometry and commutative algebra moreover grobner bases have been also generalized to free noncommutative algebra and to various noncommutative algebras of interest in differential algebra e g weyl algebras enveloping algebras of lie algebras the aim of this paper is to give an introduction as elementary as i was able to make it to both commutative and noncommutative algebras grobner bases are in a sense a finite model of an infinite linear gauss reduced basis of an ideal viewed as a vector space and buchberger algorithm is the corresponding generalization of the gaussian elimination algorithm moreover the paper contains a survey of some applications of buchberger theory to noncommutative algebras together with these results surveyed this paper contains some minor new points e g the useless pair criteria in the noncommutative case and the final result on the existence and computability of grobner bases for two sided ideals in any finitely presented algebra

the scaling limit of the minimum spanning tree of the complete graph consider the minimum spanning tree mst of the complete graph with nn vertices when edges are assigned independent random weights endow this tree with the graph distance renormalized by n1 3n1 3 and with the uniform measure on its vertices we show that the resulting space converges in distribution as n n to a random compact measured metric space in the gromov hausdorff prokhorov topology we additionally show that the limit is a random binary rr tree and has minkowski dimension 33 almost surely in particular its law is mutually singular with that of the brownian continuum random tree or any rescaled version thereof our approach relies on a coupling between the mst problem and the erd s renyi random graph we exploit the explicit description of the scaling limit of the erd s renyi random graph in the so called critical window established in probab theory related fields 152 2012 367 406 and provide a similar description of the scaling limit for a critical minimum spanning forest contained within the mst in order to accomplish this we introduce the notion of rr graphs which generalise rr trees and are of independent interest

invasion percolation on regular trees we consider invasion percolation on a rooted regular tree for the infinite cluster invaded from the root we identify the scaling behavior of its r point function for any r 2 and of its volume both at a given height and below a given height we find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation the scaling functions differ thus somewhat surprisingly the two clusters behave differently in fact we prove that their laws are mutually singular in addition we derive scaling estimates for simple random walk on the cluster starting from the root we show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster far above the root the two clusters have the same law locally but not globally a key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster these weights decay toward the critical value for ordinary percolation but only slowly and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster

stepsize restrictions for stability of one step methods in the numerical solution of initial value problems this paper deals with the analysis of general one step methods for the numerical solution of initial boundary value problems for stiff ordinary and partial differential equations restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size these restrictions are related to disks contained in the stability region of the method and the errors are measured with arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of a diffusion convection problem where the error growth is measured with the maximum norm

a step size selection strategy for explicit runge kutta methods based on lyapunov exponent theory using a stability result for variable time step runge kutta methods applied to nonautonomous real linear scalar test problem that decays exponentially fast a step size selection algorithm is devised the step size selection algorithm is based partly on stability information obtained by estimating the discrete lyapunov exponent of a runge kutta method applied to a nonautonomous linear scalar problem the utility of the approach is illustrated in numerical experiments that demonstrate how the new algorithm performs against a standard accuracy based step size selection strategy

stability of semidiscretizations of hyperbolic problems semidiscretization of hyperbolic systems yields systems of ordinary differential equations of first or second order in both cases we investigate the stability of a general class of methods namely the m stage k step methods when applied to linear test equations a map sigma is introduced which allows us at least for constant linear systems to investigate whether a second order system should be integrated directly or should first be reduced to a first order system

contractivity in the numerical solution of initial value problems consider a linear autonomous system of ordinary differential equations with the property that the norm u t of each solutionu t satisfies u t u 0 t 0 we call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations a givenk step method is said to be unconditionally contractive if for each stepsizeh 0 the numerical process is contractive in this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity it turns out that unconditionally contractive methods are subject to an order barrierp 1 further the concept of a contractivity threshold is studied which makes it possible to compare the contractivity behaviour of methods with an orderp 1 as well most theoretical results in this paper are formulated for differential equations in arbitrary banach spaces applications are given to numerical methods for solving ordinary as well as partial differential equations

an efficient algorithm based on splitting for the time integration of the schrodinger equation we present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the schrodinger equation when the hamiltonian operator is either time independent or changes slowly with time in the later case the evolution operator can be effectively approximated in a step by step manner first divide the time integration interval in sufficiently short subintervals and then successively solve a schrodinger equation with a different time independent hamiltonian operator in each of these subintervals when discretized in space the schrodinger equation with the time independent hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous hamiltonian system corresponding to a system of coupled harmonic oscillators the particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals sharp local error bounds are obtained for the solution of the linear autonomous hamiltonian system considered in each time subinterval our schemes can be considered in this setting as polynomial approximations to the matrix exponential in a similar way as methods based on chebyshev and taylor polynomials the theoretical analysis supported by numerical experiments performed for different time independent hamiltonians indicates that the new methods are more efficient than schemes based on chebyshev polynomials for all tolerances and time interval lengths the algorithm we present automatically selects for each time subinterval the most efficient splitting scheme among several new optimized splitting methods for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the hamiltonian operator

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

exponential growth of ponds in invasion percolation on regular trees in invasion percolation the edges of successively maximal weight the outlets divide the invasion cluster into a chain of ponds separated by outlets on the regular tree the ponds are shown to grow exponentially with law of large numbers central limit theorem and large deviation results the tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster with a logarithmic correction

retention capacity of random surfaces we introduce a water retention model for liquids captured on a random surface with open boundaries and investigate the model for both continuous and discrete surface heights 0 1 n 1 on a square lattice with a square boundary the model is found to have several intriguing features including a nonmonotonic dependence of the retention on the number of levels for many n the retention is counterintuitively greater than that of an n 1 level system the behavior is explained using percolation theory by mapping it to a 2 level system with variable probability results in one dimension are also found

error analysis of splitting methods for the time dependent schrodinger equation a typical procedure to integrate numerically the time dependent schrodinger equation involves two stages in the first stage one carries out a space discretization of the continuous problem this results in the linear system of differential equations i du dt h u where h is a real symmetric matrix whose solution with initial value u 0 u 0 in mathbb c n is given by u t mathrm e i t h u 0 usually this exponential matrix is expensive to evaluate so that time stepping methods to construct approximations to u from time t n to t n 1 are considered in the second phase of the procedure among them schemes involving multiplications of the matrix h with vectors such as lanczos and chebyshev methods are particularly efficient in this work we consider a particular class of splitting methods which also involves only products hu we carry out an error analysis of these integrators and propose a strategy which allows us to construct different splitting symplectic methods of different order even of order zero possessing a large stability interval that can be adapted to different space regularity conditions and different accuracy ranges of the spatial discretization the validity of the procedure and the performance of the resulting schemes are illustrated in several numerical examples

accuracy bounds for semidiscretizations of hyperbolic problems bounds are given for the error constant of stable finite difference methods for first order hyperbolic equations in one space dimension which use r downwind and s upwind points in the discretization of the space derivatives and which are of optimal order p min r s 2 r 2 2 s it is known that this order can be obtained by interpolatory methods examples show however that their error constants can be improved

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

broken ergodicity and the geometry of rugged landscapes we study the nature and mechanisms of broken ergodicity be in specific random walk models corresponding to diffusion on random potential surfaces in both one dimension and high dimension using both rigorous results and nonrigorous methods we confirm several aspects of the standard be picture and show that others apply in one dimension but need to be modified in higher dimensions these latter aspects include the notions that a fixed temperature confining barriers increase logarithmically with time that components are necessarily bounded regions of state space which depend on the observational time scale and that the system continually revisits previously traversed regions of state space we examine our results in the context of several experiments and discuss some implications of our results for the dynamics of disordered and or complex systems

skew polynomial rings with binomial relations abstract in this paper we continue the study of a class of standard finitely presented quadratic algebrasaover a fixed fieldk called binomial skew polynomial rings we consider some combinatorial properties of the set of defining relationsfand their implications for the algebraic properties ofa we impose a condition called onfand prove that in this caseais a free module of finite rank over a strictly ordered noetherian domain we show that an analogue of the diamond lemma is true for one sided ideals of a skew polynomial ringawith condition we prove also that if the set of defining relationsfis square free then condition is necessary and sufficient for the existence of a finite groebner basis of every one sided ideal ina and for left and right noetherianness ofa as a corollary we find a class of finitely generated non commutative semigroups which are left and right noetherian

constructive characterization ofa stable approximations to exp z and its connection with algebraically stable runge kutta methods all rational approximations to exp z of order 2m m denotes the maximal degree of nominator and denominator are given by a closed formula involving real parameters using the theory of order stars 9 necessary and sufficient conditions fora stability respectivelyi stability are given on the basis of this characterization relations between the concepts ofa stability and algebraic stability for implicit runge kutta methods are investigated in particular we can partly prove the conjecture that to any irreduciblea stabler z of oderp 0 there exist algebraically stable runge kutta methods of the same order withr z as stability function

stability of parallel explicit ode methods general purpose numerical methods for ordinary initial value problems are usually found in the class of linear multistage multivalue methods first formulated by j c butcher among these the explicit methods are easiest to implement for this reason there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in par allel each such group of concurrent function evaluations can be regarded as a stage of the method it is desirable that methods have a large region of stability so that stability dictated restrictions on the stepsize will occur less frequently in order to do a fair comparison of stability regions one should consider the scaled stability region this is defined using not the nominal stepsize but rather the stepsize per unit of work each stage counts as one work unit in the context of parallel computing each stage may involve any number of independent function evaluations very important results on scaled stability regions were obtained by jeltsch and nevanlinna in the context of serial computation they identify a type of method actually a type of alge braic function whose scaled stability region is optimal this means that the region is not a proper subset of some other scaled stability region the distinguished type of algebraic functions is one that satisfies property c which requires that the principal branch of the algebraic function be analytic outside the stability region except for a pole at infinity whose multiplicity equals the number of work units of the method it is shown that by modifying property c the result on optimal stability regions holds also in the context of parallel computa tion an important consequence is that although the use of parallelism enlarges the class of scaled stability regions it does not result in scaled stability regions that properly contain those obtained without parallelism roughly stated parallelism cannot improve the stability of an explicit linear method but it can even for the standard test problem improve the accuracy

the fractal volume of the two dimensional invasion percolation cluster we consider both invasion percolation and standard bernoulli bond percolation on thez2 lattice denote byv andc the invasion cluster and the occupied cluster of the origin respectively let mathcal v n mathcal v cap n n 2 and open image in new window let e 0 be given here we show that with a probability tending to 1 n 2 varepsilon pi n leqq mathcal v n leqq n 2 varepsilon pi n assuming the existence of an exponent 1 for n it can be seen that with probability tending to one n 2 1 rho varepsilon leqq mathcal v n leqq n 2 1 rho varepsilon moreover by den nijs and nienhuis et al s computations n 1 8958389583 varepsilon n 1 tfrac 43 48 varepsilon leqq mathcal v n leqq n 1 tfrac 43 48 varepsilon n 1 8958389583 varepsilon with a probability tending to one the result matches wilkinson and willemsen s numerical computation mathcal v n sim n 1 89 the method allows us also to show the same argument for any planar graph therefore any two planar invasion clusters have the same fractal dimension 2 1 if one believes universality

order stars and stiff integrators order stars introduced in g wanner e hairer s p norsett order stars and stability theorems bit 18 1978 475 489 have become a fundamental tool for the understanding of order and stability properties of numerical methods for stiff differential equations this survey retraces their discovery and their principal achievements we also sketch some later extensions and describe some recent developments

the incipient infinite cluster in two dimensional percolation letpp be the probability measure on the configurations of occupied and vacant vertices of a two dimensional graphg under which all vertices are independently occupied respectively vacant with probabilityp respectively 1 p letph be the critical probability for this system andw the occupied cluster of some fixed vertexw0 we show that for many graphsg such as mathbb z 2 or its covering graph which corresponds to bond percolation on mathbb z 2 the following two conditional probability measures converge and have the same limit v say

limit theorems for 2d invasion percolation we prove limit theorems and variance estimates for quantities related to ponds and outlets for 2d invasion percolation we first exhibit several properties of a sequence o n of outlet variables the nth of which gives the number of outlets in the box centered at the origin of side length 2n the most important of these properties describes the sequence s renewal structure and exponentially fast mixing behavior we use these to prove a central limit theorem and strong law of large numbers for o n we then show consequences of these limit theorems for the pond radii and outlet weights

computation of optimal monotonicity preserving general linear methods monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution we formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations and propose an efficient algorithm for its solution this algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and or stages the optimality of some recently proposed methods is verified and many more efficient methods are found we use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type including methods for hyperbolic pdes that use downwind biased operators