atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

metropolis monte carlo simulation of lattice animals we propose and study a metropolis monte carlo algorithm for the simulation of weighted lattice animals in in the canonical ensemble with a fixed number of edges we examine the implementation and performance of the algorithm data obtained by sampling uniformly weighted animals and animals weighted as critical percolation clusters are reported and analysed in particular we estimate autocorrelation times and dynamical exponents to study the efficiency of the algorithm and estimate the metric and branch exponents of lattice animals in two and three dimensions

the algebraic approach to the discrete cosine and sine transforms and their fast algorithms it is known that the discrete fourier transform dft used in digital signal processing can be characterized in the framework of the representation theory of algebras namely as the decomposition matrix for the regular module mathbb c z n mathbb c x x n 1 this characterization provides deep insight into the dft and can be used to derive and understand the structure of its fast algorithms in this paper we present an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of chebyshev polynomials then we derive most of their known algorithms by pure algebraic means we identify the mathematical principle behind each algorithm and give insight into its structure our results show that the connection between algebra and digital signal processing is stronger than previously understood

an optimal minimum spanning tree algorithm we establish that the algorithmic complexity of the minimumspanning tree problem is equal to its decision tree complexity specifically we present a deterministic algorithm to find aminimum spanning tree of a graph with n vertices andm edges that runs in timeo t m n wheret is the minimum number of edge weightcomparisons needed to determine the solution the algorithm isquite simple and can be implemented on a pointer machine althoughour time bound is optimal the exact function describing it is notknown at present the current best bounds known fort are t m n m and t m n o m m n where is acertain natural inverse of ackermann s function even under theassumption that t is superlinear we show thatif the input graph is selected from gn m our algorithm runs in linear time with high probability regardlessof n m or the permutation of edge weights theanalysis uses a new martingale for gn msimilar to the edge exposure martingale forgn p

a filon type asymptotic approach to solving highly oscillatory second order initial value problems abstract in this paper we analyze and design filon type asymptotic methods for solving highly oscillatory second order initial value problems q t mq t f t q t q t where m is a non singular and diagonalizable matrix having large eigenvalues and m 1 we first derive filon type asymptotic methods for highly oscillatory linear systems and then extend the methods to highly oscillatory nonlinear systems the integrators are a blend of the existing filon type methods and asymptotic methods based on the variation of constants formula it is shown that the new integrators are convergent and the accuracy of the new integrators is improved as m grows large this point is significant and powerful for effectively dealing with highly oscillatory systems numerical experiments are carried out and the numerical results show the advantage and efficiency of our new integrators as compared with existing numerical methods for highly oscillatory problems from the scientific literature

monte carlo test of dimensional reduction for branched polymers in three dimensions by a monte carlo simulation of a grand canonical ensemble for a lattice tree model of branched polymers in three dimensions the author obtains the estimates theta 1 501 or 0 043 nu 0 495 or 0 013 for the critical exponents these estimates strongly support the exact conjecture theta 3 2 nu 1 2 by parisi and sourlas 1981 obtained using dimensional reduction the attrition constant mu is also estimated

a gautschi type method for oscillatory second order differential equations we study a numerical method for second order differential equations in which high frequency oscillations are generated by a linear part for example semilinear wave equations are of this type the numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly we prove that the method admits second order error bounds which are independent of the product of the step size with the frequencies our analysis also provides new insight into the m ollified impulse method of garcia archilla sanz serna and skeel we include results of numerical experiments with the sine gordon equation

high precision canonical monte carlo determination of the growth constant of square lattice trees exponent in this paper we show that l and u can be determined to high accuracy by using a canonical monte carlo algorithm we find thatl55 143960 0025 u51 01460 022 where the error bars are a combined 95 statistical confidence interval and an estimated systematic error due to uncertainties in modeling corrections to scaling if one assumes the exact value u51 and then determines l then the above estimate improves to l55 143 3960 000 72 in addition we also determine the longest path exponent r and the metric exponent n from our data r50 7400060 00062 n50 643760 0035 with error bars similarly a combined 95 statistical confidence interval and an estimate of the systematic error

efficient energy preserving integrators for oscillatory hamiltonian systems in this paper we focus our attention on deriving and analyzing an efficient energy preserving formula for the system of nonlinear oscillatory or highly oscillatory second order differential equations q t mq t f q t where m is a symmetric positive semi definite matrix with m 1 and f q q u q is the negative gradient of a real valued function u q this system is a hamiltonian system with the hamiltonian h p q 1 2 p t p 1 2 q t mq u q the energy preserving formula exactly preserves the hamiltonian we analyze in detail the properties of the energy preserving formula and propose new efficient energy preserving integrators in the sense of numerical implementation the convergence analysis of the fixed point iteration is presented for the implicit integrators proposed in this paper it is shown that the convergence of implicit average vector field methods is dependent on m whereas the convergence of the new energy preserving integrators is independent of m the fermi pasta ulam problem and the sine gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well known average vector field methods in the scientific literature

enumerations of lattice animals and trees we have developed an improved algorithm that allows us to enumerate the number of site animals on the square lattice up to size 46 we also calculate the number of lattice trees up to size 44 and the radius of gyration of both lattice animals and trees up to size 42 analysis of the resulting series yields an improved estimate 4 062570 8 for the growth constant of lattice animals and 0 3 795254 8 for the growth constant of trees and confirms to a very high degree of certainty that both the animal and tree generating functions have a logarithmic divergence analysis of the radius of gyration series yields the estimate 0 64115 5 for the size exponent

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

asymptotic expansion and quadrature of composite highly oscillatory integrals we consider in this paper asymptotic and numerical aspects of highly oscillatory integrals of the form b a f x g sin x dx where w 1 such integrals occur in the simulation of electronic circuits but they are also of independent mathematical interest the integral is expanded in asymptotic series in inverse powers of this expansion clarifies the behaviour for large w and also provides a powerful means to design effective computational algorithms in particular we introduce and analyse filon type methods for this integral

efficient computation of the matrix exponential by generalized polar decompositions in this paper we explore the computation of the matrix exponential in a manner that is consistent with lie group structure our point of departure is the method of generalized polar decompositions which we modify and combine with similarity transformations that bring the underlying matrix to a form more amenable to efficient computation we develop techniques valid for a range of lie groups the orthogonal group the symplectic group lorentz isotropy and scaling groups however the gpd approach is equally promising in a more general context even when lie group structure is not at issue our algorithm is more efficient in many settings than classical methods for the computation of the matrix exponential

the collapse transition for the lattice trees the authors consider a lattice tree model for the collapse of dilute branched polymers in the good solvent regime in which the collapse is driven by a near neighbour contact fugacity they describe some rigorous results including bounds for the temperature dependence of the reduced limiting free energy and compare these results with numerical estimates derived from exact enumeration data from the specific heat they estimate the collapse temperature tc and the cross over exponent phi 0 in two and three dimensions they find phi 0 0 60 or 0 03 d 2 and phi 0 0 82 or 0 03 d 3 finally they speculate on a possible roughening transition which may occur at a temperature tr tc

on sor waveform relaxation methods waveform relaxation is a numerical method for solving large scale systems of ordinary differential equations on parallel computers it differs from standard iterative methods in that it computes the solution on many time levels or along a continuous time interval simultaneously this paper deals with the acceleration of the standard waveform relaxation method by successive overrelaxation sor techniques in particular different sor acceleration schemes based on multiplication with a scalar parameter or convolution with a time dependent function are described and theoretically analyzed the theory is applied to a one dimensional and two dimensional model problem and checked against results obtained by numerical experiments

long time energy conservation of numerical methods for oscillatory differential equations we consider second order differential systems where high frequency oscillations are generated by a linear part we present a frequency expansion of the solution and we discuss two invariants of the system that determine the coefficients of the frequency expansion these invariants are related to the total energy and the oscillatory harmonic energy of the original system for the numerical solution we study a class of symmetric methods that discretize the linear part without error we are interested in the case where the product of the step size with the highest frequency can be large in the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system this allows us to prove the near conservation of the total and the oscillatory energy over very long time intervals

explicit runge kutta nystro m methods with reduced phase errors for computing oscillating solutions we construct explicit runge kutta nystrom methods for the integration of first and second order differential equations having an oscillatory solution special attention is paid to the phase errors or dispersion of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model rk n methods are constructed which are dispersive of orders up to 10 whereas the algebraic order of accuracy is only 2 or 3 application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced

unusual lienard type nonlinear oscillator a lienard type nonlinear oscillator of the form x kxx k2 9 x3 lambda1x 0 which may also be considered as a generalized emden type equation is shown to possess unusual nonlinear dynamical properties it is shown to admit explicit nonisolated periodic orbits of conservative hamiltonian type for lambda1 0 these periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator this is completely contrary to the standard characteristic property of nonlinear oscillators interestingly the system though appears deceptively a dissipative type for lambda1 or 0 does admit a conserved hamiltonian description where the characteristic decay time is also independent of the amplitude the results also show that the criterion for conservative hamiltonian system in terms of divergence of flow function needs to be generalized

explicit multi symplectic extended leap frog methods for hamiltonian wave equations in this paper we study the integration of hamiltonian wave equations whose solutions have oscillatory behaviors in time and or space we are mainly concerned with the research for multi symplectic extended runge kutta nystrom erkn discretizations and the corresponding discrete conservation laws we first show that the discretizations to the hamiltonian wave equations using two symplectic erkn methods in space and time respectively lead to an explicit multi symplectic integrator eleap frogi then we derive another multi symplectic discretization using a symplectic erkn method in time and a symplectic partitioned runge kutta method which is equivalent to the well known stormer verlet method in space eleap frogii these two new multi symplectic schemes are extensions of the leap frog method the numerical stability and dispersive properties of the new schemes are analyzed numerical experiments with comparisons are presented where the two new explicit multi symplectic methods and the leap frog method are applied to the linear wave equation and the sine gordon equation the numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation

a rigorous bound on the critical exponent for the number of lattice trees animals and polygons the number ofn site lattice trees up to translation is believed to behave asymptotically ascn 0 n where is a critical exponent dependent only on the dimensiond of the lattice we present a rigorous proof that d 1 d for anyd 2 the method also applies to lattice animals site animals and two dimensional self avoiding polygons we also prove that v whend 2 wherev is the exponent for the radius of gyration

rooted spiral trees on hyper cubic lattices we study rooted spiral trees in 2 3 and 4 dimensions on a hyper cubic lattice using exact enumeration and monte carlo techniques on the square lattice we also obtain exact lower bound of 1 93565 on the growth constant series expansions give 1 3667 0 0010 and 0 6574 0 0010 on a square lattice with monte carlo simulations we get the estimates as 1 364 0 010 and 0 656 0 010 these results are numerical evidence against earlier proposed dimensional reduction by four in this problem in dimensions higher than two the spiral constraint can be implemented in two ways in either case our series expansion results do not support the proposed dimensional reduction

generalized atmospheric rosenbluth methods garm we show that the classical rosenbluth method for sampling self avoiding walks hammersley and morton 1954 j r stat soc b 16 23 rosenbluth and rosenbluth 1955 j chem phys 23 356 can be extended to a general algorithm for sampling many families of objects including self avoiding polygons the implementation relies on an elementary move which is a generalization of kinetic growth rather than only appending edges to the endpoint edges may be inserted at any vertex provided the resulting objects still lie within the same family this gives for the first time a kinetic growth algorithm for sampling self avoiding polygons we implement this method using pruning and enrichment grassberger 1997 phys rev e 56 3682 to sample self avoiding walks and polygons the algorithm can be further extended by mixing it with length preserving moves such as pivots and crank shaft moves

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

new lower bounds on the self avoiding walk connective constant we give an elementary new method for obtaining rigorous lower bounds on the connective constant for self avoiding walks on the hypercubic lattice d the method is based on loop erasure and restoration and does not require exact enumeration data our bounds are best for highd and in fact agree with the first four terms of the 1 d expansion for the connective constant the bounds are the best to date for dimensionsd 3 but do not produce good results in two dimensions ford 3 4 5 and 6 respectively our lower bound is within 2 4 0 43 0 12 and 0 044 of the value estimated by series extrapolation

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

simulations of lattice animals and trees the scaling behaviour of randomly branched polymers in a good solvent is studied in two to nine dimensions using as microscopic models lattice animals and lattice trees on simple hypercubic lattices as a stochastic sampling method we use a biased sequential sampling algorithm with re sampling similar to the pruned enriched rosenbluth method perm used extensively for linear polymers essentially we start simulating percolation clusters either site or bond re weigh them according to the animal tree ensemble and prune or branch the further growth according to a heuristic fitness function in contrast to previous applications of perm this fitness function is not the weight with which the actual configuration would contribute to the partition sum but is closely related to it we obtain high statistics of animals with up to several thousand sites in all dimension 2 d 9 in addition to the partition sum number of different animals we estimate gyration radii and numbers of perimeter sites in all dimensions we verify the parisi sourlas prediction and we verify all exactly known critical exponents in dimensions 2 3 4 and 8 in addition we present the hitherto most precise estimates for growth constants in d 3 for clusters with one site attached to an attractive surface we verify for d 3 the superuniversality of the cross over exponent at the adsorption transition predicted by janssen and lyssy but not for d 2 there we find 0 480 4 instead of the conjectured 1 2 finally we discuss the collapse of animals and trees arguing that our present version of the algorithm is also efficient for some of the models studied in this context but showing that it is not very efficient for the classical model for collapsing animals

efficiency of the incomplete enumeration algorithm for monte carlo simulation of linear and branched polymers we study the efficiency of the incomplete enumeration algorithm for linear and branched polymers there is a qualitative difference in the efficiency in these two cases the average time to generate an independent sample of configuration of polymer with n monomers varies as n2 for linear polymers for large n but as exp cn for branched undirected and directed polymers where 0 1 on the binary tree our numerical studies for n of order 104 gives 0 333 0 005 we argue that 1 3 exactly in this case

the scaling and squaring method for the matrix exponential revisited the scaling and squaring method is the most widely used method for computing the matrix exponential not least because it is the method implemented in the matlab function expm the method scales the matrix by a power of 2 to reduce the norm to order 1 computes a pade approximant to the matrix exponential and then repeatedly squares to undo the effect of the scaling we give a new backward error analysis of the method in exact arithmetic that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency we also give a new rounding error analysis that shows the computed pade approximant of the scaled matrix to be highly accurate for ieee double precision arithmetic the best choice of degree of pade approximant turns out to be 13 rather than the 6 or 8 used by previous authors our implementation of the scaling and squaring method always requires at least two fewer matrix multiplications than the expm function in matlab 7 0 when the matrix norm exceeds 1 which can amount to a 37 saving in the number of multiplications and it is typically more accurate owing to the fewer required squarings we also investigate a different scaling and squaring algorithm proposed by najfeld and havel that employs a pade approximation to the function x coth x this method is found to be essentially a variation of the standard one with weaker supporting error analysis