the decomposition method applied to systems of fredholm integral equations of the second kind in this paper the adomian decomposition method is applied to solve systems of linear and nonlinear fredholm integral equations of the second kind convergence of the method is proved and some examples are presented to illustrate the method

shadowing orbits of ordinary differential equations abstract a new notion of shadowing of a pseudo orbit an approximate solution of an autonomous system of ordinary differential equations by an associated nearby true orbit is introduced then a general shadowing theorem for finite time which guarantees the existence of shadowing in ordinary differential equations and provides error bounds for the distance between the true and the pseudo orbit in terms of computable quantities is proved the use of this theorem in numerical computations of orbits is outlined

perturbation expansions of the axial next nearest neighbour ising and asymmetric four state clock models the axial next nearest neighbour ising annni and four state asymmetric clock 4 asc models are investigated by perturbing about points at which they reduce the simple nearest neighbour ising models these limits are a for the annni model i small nearest neighbour interaction j1 in the x direction and ii small next nearest neighbour interaction j2 in the x direction and b for the 4 asc model small asymmetry delta in the limit j1 to 0 the phase boundary is located to first order in j1 and found to be ising like in the limit j2 to 0 the behaviour to second order in j2 is more complicated but is identical to that found to second order in delta for the 4 asc model these results are discussed in the light of other recent work

conformal invariance and the critical behavior of the triplet xy quantum chain we introduce and study the critical properties of the tripletxy quantum chain this system is described in terms of three spin interactions and is the generalization of the standardxy quantum chain we show that this model with periodic boundaries has a local gauge invariance and can be described by the composition of two triplet ising models with general toroidal boundary conditions from this composition the phase diagram as well the conformai anomaly and critical exponents are determined by exploring their relations with the mass gap amplitudes predicted by conformai invariance

shadowing orbits of ordinary differential equations on invariant submanifolds a finite time shadowing theorem for autonomous ordinary differential equations is presented under consideration is the case were there exists a twice continuously differentiable function g mapping phase space into lrtm with the property that for a particular regular value c of g the submanifold g 1 c is invariant under the flow the main theorem gives a condition which implies that an approximate solution lying close to g1 c is uniformly close to a true solution lying in g 1 c applications of this theorem to computer generated approximate orbits are discussed

conformal anomaly and critical exponents of the spin 1 takhtajan babujian model the authors consider the critical behaviour of the spin 1 hamiltonian of the takhtajan babujian model the bethe ansatz equations for this antiferromagnet in a finite chain are investigated analytically and numerically the conformal anomaly and critical exponents are obtained by exploiting their relations with the eigenspectrum of the finite system the results strongly support the conjecture that the wess zumino witten non linear sigma model with topological charge k 2 is the underlying field theory for this statistical mechanics model

second kind integral equations for the classical potential theory on open surfaces ii a second kind integral equation formulation is presented for the dirichlet problem for the laplace equation in two dimensions with the boundary conditions specified on a collection of open curves the performance of the obtained apparatus is illustrated with several numerical examples the formulation is a simplification of the equation previously constructed by the authors

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

on the solution of linear and nonlinear integral equation the purpose of this paper is to establish the solution of fredholm volterra integral equation of the second kind in the space l 2 w xc 0 t considering the following when fredholm integral term in li 2 w and volterra integral term in 0 t are linear also each of fredholm or volterra integral term is linear while the other term is nonlinear here w is defined as the domain of integration with respect to position while t is consider as the time email protected 0 t t also the fredholm volterra integral equation of the first kind is established as special case of these work many special in one two and three dimensional problems of integral equation are considered representing the kernel of fredholm integral term in the form of logarithmic function carleman function potential and generalized potential function and macdonald function are considered

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

the yang lee edge singularity by the phenomenological renormalisation group the phenomenological renormalisation group is applied to the 1d transverse ising model with longitudinal imaginary field in order to investigate the yang lee edge singularity 1952 of the equivalent 2d ising model the method is extended to calculate directly the exponent sigma of the magnetisation

a new algorithm for symbolic integral with application in this paper by using the theories and methods of mathematical analysis and computer algebra a new algorithm of symbolic integral was established a new maple procedure myint for it was established too and it was applied to the automatic proving the mean value theorems for derivatives the results indicated that the algorithm of symbolic integral had been established in this paper had advantage of simple idea excellent property for operation and powerful competence this would be useful for the problem of solving differential equations automatic proving some mathematical theorems and so on

numerical analysis of graded mesh methods for a class of second kind integral equations on the real line abstract in this paper we are concerned with the numerical analysis of the collocation method based on graded meshes of second kind integral equations on the real line of the form s s r s t z t t dt s r where l 1 r z l r and bc r the space of bounded continuous complex valued functions on r are assumed known and the function bc r is to be determined we introduce some new graded meshes for the collocation method of the integral equation which are different from those used previously for the wiener hopf integral equation in the case when the solution decays exponentially at infinity and establish optimal local and global l norm error estimates under the condition that the solution decays only polynomially at infinity

phase diagram of the icosahedral and dodecahedral vector models in two dimensions fixed points of discrete vector models with icosahedral and dodecahedral symmetry are found by approximate real space renormalisation techniques both models have three distinct phases similar to cubic vector models new fixed points governing the order disorder and partial order disorder transitions are identified these transitions gradually become softer as the number of vector components increases

a shadowing lemma approach to global error analysis for initial value odes the authors show that for dynamical systems that possess a type of piecewise hyperbolicity in which there is no decrease in the number of stable modes the global error in a numerical approximation may be obtained as a reasonable magnification of the local error in particular under certain conditions the authors prove the existence of a trajectory on an infinite time interval of the given ordinary differential equation uniformly close to a given numerically computed orbit of the same differential equation by allowing for different initial conditions for finite time intervals a general result is proved for obtaining a posteriori bounds on the global error based on computable quantities and on finding and bounding the norm of a right inverse of a particular matrix two methods for finding and bounding estimating the norm of a right inverse are considered one method is based upon the choice of the pseudo or generalized inverse the other method is based upon solving multipoint boundary value problems bv

subharmonic solutions of systems of difference equations with periodic perturbations part i existence we consider the existence of subharmonic solutions of systems of difference equations with periodic perturbations the theory of coincidence degree coupled with some detailed a priori estimates is applied to show that the perturbed system admits subharmonic solutions near a hyperbolic periodic orbit of the unperturbed system if the perturbation is sufficiently small

the 3 state potts model in 2 1 dimensions the 3 state potts model in 2 1 dimensions is studied using series and monte carlo methods high temperature series for the vacuum energy mass gap and susceptibility are computed on the square and triangular lattices using linked cluster expansions monte carlo estimates are obtained by the stochastic truncation method the model is found to undergo a weak first order transition as in the euclidean version of the model

finite size scaling and universality in the 2 1 d ising model the energy and magnetisation density of the 2 1 d ising model are investigated for the thermal exponent the author finds xepsilon 1 42 or 0 02 in agreement with hyperscaling the privman fisher universality hypothesis is confirmed by two methods first the author presents at the critical point numerical data for the finite size scaling amplitudes of the singular free energy magnetisation susceptibility and surface tension and checks them for universality second using the explicitly computed eigenvectors of the hamiltonian it is shown that the ratios of matrix elements of the energy and magnetisation density are universal

a modified approach to numerical solution of fredholm integral equations of the second kind a modified approach to obtain approximate numerical solutions of fredholm integral equations of the second kind is presented the error bound is explained by the aid of several illustrative examples in each example the approximate solution is compared with the exact solution wherever possible and an excellent agreement is observed in addition the error bound in each example is compared with the one obtained by the nystrom method it is found that the error bound of the present method is smaller than the ones obtained by the nystrom method further the present method is successfully applied to derive the solution of an integral equation arising in a special dirichlet problem

crumpling transition of the triangular lattice without open edges effect of a modified folding rule folding of the triangular lattice in a discrete three dimensional space is investigated by means of the transfer matrix method this model was introduced by bowick and co workers as a discretized version of the polymerized membrane in thermal equilibrium the folding rule constraint is incompatible with the periodic boundary condition and the simulation has been made under the open boundary condition in this paper we propose a modified constraint which is compatible with the periodic boundary condition technically the restoration of translational invariance leads to a substantial reduction in the transfer matrix size treating the cluster sizes l ensuremath le 7 we analyze the singularities of the crumpling transitions for a wide range of the bending rigidity k we observe a series of the crumpling transitions at k 0 206 2 ensuremath 0 32 1 and ensuremath 0 76 10 at each transition point we estimate the latent heat as q 0 356 30 0 08 3 and 0 05 5 respectively

numerical shadowing near hyperbolic trajectories shadowing is a means of characterizing global errors in the numerical solution of initial value ordinary differential equations by allowing for a small perturbation in the initial condition the method presented in this paper allows for a perturbation in the initial condition and a reparameterization of time in order to compute the shadowing distance in the neighborhood of a periodic orbit or more generally in the neighborhood of an attractor the method is formulated for one step methods and both a serial and parallel implementation are applied to the forced van der pol equation the lorenz equation and to the approximation of a periodic orbit

the behavior of finite element solutions of semilinear parabolic problems near stationary points the qualitative behavior of spatially semidiscrete finite element solutions of a semilinear parabolic problem near an unstable hyperbolic equilibrium overline u is studied it is shown that any continuous trajectory is approximated by an appropriate discrete trajectory and vice versa as long as they remain in a sufficiently small neighborhood of overline u error bounds of optimal order in the l 2 and h 1 norms hold uniformly over arbitrarily long time intervals in particular the local stable and unstable manifolds of the discrete problem converge to their continuous counterparts therefore the discretized dynamical system has the same qualitative behavior near overline u as the continuous system

stochastic truncation for the 2 1 d ising model the method of symmetrized stochastic truncation is applied to the 2 1 dimensional ising model to calculate the ground state energy mass gap magnetization and susceptibility on lattices of sizes up to 6 6 sites the method is shown to be a powerful and accurate monte carlo technique provided the lattice size is not too large and the model is suitably convergent finite size scaling estimates of the critical indices are nu 0 627 2 alpha 0 12 2 beta 0 324 3 and delta 1 23 1 an estimate of the combination gamma nu 2 beta nu 3 000 1 is obtained in outstanding agreement with the expected hyperscaling relation

long range one dimensional potts models a cluster mean field and extrapolation approach we obtain approximations for the critical temperature tc of q state ferromagnetic potts models on one dimensional lattices with algebraically decaying ferromagnetic pair interactions i e decaying as 1 r with 1 2 initially we use a cluster mean field method to get approximations to the critical temperature and we look at increasingly large clusters this gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms we see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate tc estimates our best estimates we believe to be correct to four figure accuracy

finite size scaling study of a first order temperature driven symmetry breaking structural phase transition we present a study of finite size effects in a model exhibiting a first order temperature driven symmetry breaking structural phase transition in thel cylindrical geometry in thel limit exact studies demonstrate the applicability of our scaling ansatz even in the one dimensional limit making this model ideal for studying finite size effects the scaling ansatz similar to the previously developed ansatz for field driven transitions demonstrates that latent heat is crucial in driving these transitions this ansatz is supported by a 2 2 phenomenological transfer matrix based upon the symmetries of the system this produces an analytic free energy which has the scaling form order parameter probability distributions show that the high and low temperature phases coexist only in a small finite size affected regime near the bulk transition temperature this regime vanishes exponentially fast asl diverges

finite range scaling study of the 1d long range ising model the critical behaviour of the one dimensional ising model with long range ferromagnetic interactions decaying with distance r as 1 r1 sigma has been studied by scaling the range of interactions exact calculations have been done for a system with finite ranges up to the tenth neighbour for 0 sigma or 1 applying the range scaling the critical temperature critical exponent nu and the anomalous dimension of the order parameter have been calculated additional analysis of the convergence of the method has been performed by applying the vanden broeck and schwartz extrapolation procedure in addition to the simple least squares approximation and by evaluating the convergence exponents

approximate and real trajectories for generic dynamical systems abstract in this paper we show that generic dynamical systems have a weak shadowing property namely that for a generic dynamical system in the space of discrete continuous dynamical systems on a compact manifold m with the usual c 0 topology given 0 there exists a 0 such that trajectories are close to real trajectories there are no dimensional restrictions on this result we also give some results on the inverse problem of tracing real trajectories by trajectories these problems are motivated by questions regarding the validity of numerical solutions of differential equations

a simplification to fredholm s solution to the fredholm integral equation of the second kind abstract fredholm s classic solution to his linear integral equation of the second kind has been generally considered to be too complex to provide a practical solution method it gives the resolvent kernel as a ratio of two infinite series with terms in each series containing multiple integrals of determinants such that the evaluation of the nth term would seem to require n different n fold multi integrations a formidable task even for modest values of n this paper shows however that by using the methods of the combinatorics of permutations and by solving a well known recursion relation fredholm s solution can be simplified considerably in this simplified form the nth term of each series requires the evaluation of just one new multi integral thus presented is a new expression for the fredholm s solution and the practical use of this expression is demonstrated on some example problems

mean field renormalization group for the boundary magnetization of strip clusters we analyse in some detail a recently proposed transfer matrix mean field approximation which yields the exact critical point for several two dimensional nearest neighbour ising models for the square lattice model we show explicitly that this approximation yields not only the exact critical point but also the exact boundary magnetization of a semi infinite ising model independent of the size of the strips used then we develop a new mean field renormalization group strategy based on this approximation and make connections with finite size scaling applying our strategy to the quadratic ising and three state potts models we obtain results for the critical exponents which are in very good agreement with the exact ones in this way we also clarify some advantages and limitations of the mean field renormalization group approach

on subharmonic solutions of systems of difference equations with periodic perturbations part ii multiplicity and stability we consider the multiplicity and stability of subharmonic solutions of discrete dynamic systems with periodic perturbations whose existence was established in the first part of this series under the hypotheses that the perturbation is locally lipschitz continuous we determine the precise number of subharmonic solutions and we also show that these subharmonic solutions are stable unstable provide the periodic orbit of the unperturbed system is stable unstable

magnetisations from finite size scaling a method of obtaining the spontaneous magnetisation from finite lattice matrix elements of the magnetic field operator due to yang 1952 and uzelac 1980 is discussed the method is demonstrated for the case of the ising model in 1 1 dimensions and is shown to provide smooth and rapidly convergent finite lattice sequences applied to the case of the three state potts z3 model in 1 1 dimensions the method yields an estimate beta 0 111 09 or 0 000 05 for the critical exponent this confirms alexander s conjecture of universality with the hard hexagonal model

finite size corrections for ground states of the xxz heisenberg chain the method of de vega and woynarovitch 1985 is used to calculate finite size corrections to the ground state energy in different sectors for the xxz heisenberg chain finite size scaling amplitudes and correction to scaling exponents in the critical region are derived using conformal invariance a scaling dimension x pi gamma 2 pi is extracted corresponding to the electric field operator in the 8 vertex model this confirms a conjecture of baxter and kelland 1974 finite size scaling properties near the kosterlitz thouless critical point delta 1 are discussed

a numerical c 1 shadowing result for retarded functional differential equations the aim of the present paper is to give a numerical c1 shadowing between the exact solutions of a functional differential equation and its numerical approximations the shadowing result is obtained by comparing exact solutions with numerical approximation which do not share the same initial value behavior of stable manifolds of functional differential equations under numerics will follow from the shadowing result

critical behaviour of the 1d q state potts model with long range interactions the critical behaviour of the one dimensional q state potts model with long range interactions decaying with distance r as r 1 sigma has been studied in the wide range of parameters 0 sigma or 1 and 1 16 or q or 64 a transfer matrix has been constructed for a truncated range of interactions for integer and continuous q and finite range scaling has been applied results for the phase diagram and the correlation length critical exponent are presented

an algorithm for solving daes with mechanization in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of pade approximation method for solving differential algebraic equation systems was established and a new maple procedures mainproc was established too ten examples are presented to illustrate the implementation of the program

an algorithm for solving nonlinear singular perturbation problems with mechanization in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of pade approximation method for solving nonlinear singular perturbation problems was established a new maple procedure spp was established too the procedure spp give not only the approximate analytic solutions but also the numerical solutions of spp six examples are presented to illustrate the implementation of the program the method is very simple and effective for most of singular perturbation problems this would be useful for solving differential equations

yang lee edge singularity in the one dimensional long range ising model the finite range scaling method has been used to study the yang lee edge singularity problem for the one dimensional ferromagnetic ising model with long range interactions decaying with distance r as r d sigma the phase diagram and the critical exponent nu have been calculated for different values of sigma by using finite ranges l or 9 the critical value sigma c dividing the long range from the short range critical regime has been discussed

finite lattice methods in quantum hamiltonian field theory i o 2 and o 3 heisenberg models for pt i see ibid vol 14 p 241 57 1981 two efficient methods for finding the low lying states of hamiltonians on finite lattices are described the first involves constructing a finite representation of the hamiltonian using strong coupling eigenstates while the second is based on the lanczos recursion method the methods are used to determine the mass gap of the o 2 and o 3 heisenberg hamiltonians in 1 1 dimensions for a sequence of finite chains the critical behaviour of the infinite chain is then analysed by extrapolating the finite lattice estimates using finite size scaling a remarkably sensitive test is developed for the presence of a phase transition for the o 2 model data this test yields strong evidence for a phase transition with the weak coupling phase massless while in the o 3 case the test supports although more weakly the absence of any transition

numerical study of conformal invariance in rsos models the author studies conformal invariance in the rsos models of andrews et al 1985 at both the ferromagnetic and antiferromagnetic transitions using finite size scaling of the transfer matrix the author confirms the conjecture of huse 1984 concerning the conformal classification of the ferromagnetic transitions and gives evidence that the antiferromagnetic transitions lie in the universality class of zn models in addition he shows that conformal results can be applied to anisotropic systems if logarithmic corrections are taken into account

transfer matrix study of the chiral clock model in the hamiltonian limit the logarithm of the transfer matrix of the two dimensional three state chiral clock model is shown to be equivalent to a one dimensional non hermitian quantum hamiltonian in the region of small chirality the critical behaviour of the quantum model is investigated by a series expansion and by finite size scaling a single phase boundary between a modulated paramagnetic and a ferromagnetic phase is located with good accuracy both methods yield comparable results for the critical exponent beta of the wavevector of the modulated high temperature phase the critical exponents nu x nu tau for the correlation lengths obtained from finite size scaling point towards the existence of a lifshitz point at finite chirality the results of the series expansion for the exponent nu x however are inconsistent with finite size scaling for larger chirality the authors finite size results clearly reveal the occurrence of an incommensurate phase with algebraically decaying correlations between the ferromagnetic and the modulated paramagnetic phase they analyse the critical behaviour of this incommensurate phase at its melting line and determine the exponent eta that controls the algebraic decay of correlations

mean field approximation with coherent anomaly method for a non equilibrium model a mean field approximation for a 1d non equilibrium lattice model known as model a is considered allowing the authors to compute the probability of a configuration of a given cluster of sites the mean field equations are numerically solved in the stationary regime for different cluster sizes l or 15 from a finite size analysis and the application of the coherent anomaly method cam they compute the critical parameter of the model the correlation length critical exponent nu perpendicular to and the order parameter critical exponent beta the results are in agreement with those obtained by other methods

using the wpg method for solving integral equations of the second kind abstract in this paper we use the wavelet petrov galerkin wpg method based on discontinuous orthogonal multiwavelets for solving fredholm integral equations of the second kind that yields linear system that the condition number of its coefficient matrix is bounded and for showing efficiency of method we use numerical examples

finite size scaling in the 2 1 d ising model hamiltonian eigenvalues are calculated for the 2 1 d ising model on lattices up to 5 5 sites finite size scaling techniques are used to estimate the critical coupling xc 0 329 or 0 001 and index v 0 635 or 0 005 the prospects are discussed for further applications of these methods in three and four dimensions

conformal invariance and correction to finite size scaling applications to the three state potts model corrections to finite size scaling are determined numerically for several levels of the three state potts quantum chain with various boundary conditions it is found that the leading correction term behaves like wo in the case of periodic and twisted boundary conditions the coefficients of the n o term are determined by the three point correlation functions of the conformal theory

a new mechanical algorithm for solving the second kind of fredholm integral equation integral equation has been one of principal tool of various areas of applied mathematics physics and engineering in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm for solving the second kind of fredholm integral equation is established and a mechanical maple algorithm fredholmproc is established too the algorithm can provide the continuous solution of the second kind of fredholm integral equation ten examples are presented to illustrate the efficiency and accuracy of the algorithm this will be useful for solving the integral equation

critical behaviour of mixed heisenberg chains the critical behaviour of anisotropic heisenberg models with two kinds of antiferromagnetically exchange coupled centres are studied numerically by using finite size calculations and conformal invariance these models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism most of our results are centred in the mixed heisenberg chain where we have at even odd sites a spin su 2 operator interacting with a xxz like interaction anisotropy our results indicate universal properties for all these chains the whole phase where the models change from ferromagnetic to ferrimagnetic behaviour is critical along this phase the critical fluctuations are ruled by a c 1 conformal field theory of gaussian type the conformal dimensions and critical exponents along this phase are calculated by studying these models with several boundary conditions

an algorithm for adomian decomposition method in this paper a reliable technique for calculating adomian polynomials for nonlinear operators is developed the algorithm is fulfilled with symbolic computation so that the adomian decomposition methods for solving differential equations can be mechanized in maple ten examples are presented to illustrate the implementation of the program

on structural stability for semiflows some results on structural stability which are known to hold for flows of a compact manifold are extended to semiflows of a banach space

finite size scaling studies of the o 2 heisenberg spin model in d 1 dimensions the finite size scaling behaviour of the o 2 heisenberg spin model in d 1 dimensions d 1 and 2 is computed in the low temperature region using the stochastic truncation monte carlo method the numerical results so obtained agree very well with effective lagrangian theory and with spin wave perturbation theory

the critical behaviour of self dual z n spin systems finite size scaling and conformal invariance this paper is concerned with the critical properties of a family of self dual two dimensional z n models whose bulk free energy is exactly known at the self dual point the author s analysis is performed by studying the finite size behaviour of the corresponding one dimensional quantum hamiltonians which also possess an exact solution at their self dual point by exploring finite size scaling ideas and the conformal invariance of the critical infinite system the author calculates for n up to 8 the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra the results strongly suggest that the z n quantum field theory of zamolodchikov and fateev is the underlying field theory associated with these statistical mechanical systems the author also tests for the z 5 case the conjecture that these models correspond to the bifurcation points in the phase diagram of the general z n spin model where a massless phase originates

finite size scaling and conformal invariance in a self dual quantum z 5 model considers the critical behaviour of a particular set of onedimensional self dual models with z n symmetry n or 5 the critical indices are evaluated using standard finite size scaling and by exploiting their relations with the mass gap amplitudes predicted by conformal invariance the results strongly suggest that the recently introduced z n quantum field theory is the underlying field theory for these statistical mechanics models

an investigation into integral equation methods involving nearly singular kernels for acoustic scattering this work presents a numerical means of investigating the acoustic scattering by disk shaped bodies on the basis of burton and miller s method the singularity free formulations of helmholtz integral equation and its normal derivative are used to form a composite equation a triangle polar co ordinate transformation method is further applied to treat the nearly singular kernels arising from a situation in which the field points and source points are very close together numerical simulations consist of the acoustic scattering by a short circular cylinder and a thin circular disk respectively for the latter case with zero thickness the corresponding analytical solutions involving angular and radial oblate spheroidal wave functions are evaluated as well comparing the numerical results with the experimental data and analytical solutions demonstrates the effectiveness of the proposed method

rigorous computational shadowing of orbits of ordinary differential equations the existence of a true orbit near a numerically computed approximate orbit shadowing of autonomous system of ordinary differential equations is investigated a general shadowing theorem for finite time which guarantees the existence of shadowing in ordinary differential equations and provides error bounds for the distance between the true and the approximate orbit in terms of computable quantities is proved the practical use and the effectiveness of this theorem is demonstrated in the numerical computations of chaotic orbits of the lorenz equations

continuously varying exponents in magnetic hard squares the commuting row to row transfer matrices of magnetic hard squares on the multicritical t manifold are shown to satisfy a special functional equation called an inversion identity for strip widths up to n 32 these equations are solved numerically for the transfer matrix eigenvalues the central charge or conformal anomaly c 1 is obtained from 1 n2 corrections to the exact bulk free energy and scaling dimensions are obtained from the eigenvalue gaps the magnetic scaling dimension is xm 1 8 and the sublattice density difference scaling dimension is xe 1 9 2 y where y 2 lambda pi and the interaction dependent crossing parameter lambda varies between 0 and 2 pi 3 scaling dimensions for further operators fall in the sequence xn n2xe where n 1 2 3 4 the critical exponents also vary continuously along the multicritical line and are given by alpha 14 9y 16 9y beta m 9 2 y 16 16 9y beta e 1 2 16 9y nu 9 2 y 2 16 9y and delta 15 these exponents are simply related to the exponents of the critical eight vertex and ashkin teller models which also exhibit z2 z2 symmetry and the authors argue that these models lie in the same universality class

approximate universality at a first order transition the three state potts model in 3 1 dimensions both high and low temperature perturbation series are used to locate and characterize the first order transition in the three state potts model in 3 1 dimensions on the simple cubic body centred cubic and face centred cubic lattices estimates are presented for the vacuum energy latent heat magnetization susceptibility and mass gap at the transition the results show a remarkable degree of universality between the different lattices and a law of corresponding states is followed very closely

numerical solution of non linear fredholm integral equations by using multiwavelets in the petrov galerkin method in this paper we use the wavelet petrov galerkin wpg method for solving one of the most important cases in non linear fredholm integral equations which is called hammerstein equations we suppose that the trial space and the test space that are used in this method to be piecewise polynomials space also we use alpert s multiwavelets as the bases for these spaces such that the wavelet basis for the test space constructed from lower degree polynomials than the trial space finally for showing efficiency of this method we use some numerical examples

q state potts models in hamiltonian field theory for q 4 in 1 1 dimensions finite lattice sequence extrapolation methods are applied to the q state potts models in 1 1 d hamiltonian field theory thermal exponents nu 0 71 or 0 02 alpha 0 53 or 0 02 are obtained for the four state model methods are discussed for treating the first order transitions for q 4 and a sequence extrapolation is used to estimate the latent heat exact analytic formulae are derived for the ground state energy the mass gap and the latent heat at the transition point these are compared with the numerical results

mechanization for solving spp by reducing order method in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of reduction of order approximation method for solving singular perturbation problems was established a new maple procedure redordproc was established too the procedure redordproc give not only the approximate analytic solutions but also the numerical solutions of the kinetics problems some examples are presented to illustrate the implementation of the program

approximation schemes for finite lattices two algorithms are formulated for systematically truncating the set of basis states required in lattice hamiltonian calculations thus allowing larger lattice abort sizes to be treated they are tested for the case of the 1 1 dimensional ising model it is shown that smoothly convergent finite lattice sequences are obtained which can successfully the extrapolated to the bulk limit the authors point out the relevance of the technique for lattice calculations in higher dimensions

a new nystrom type method for fredholm integral equations of the second kind a new nystrom type method is proposed in the paper to solve linear fredholm integral equations of the second kind convergence and error analysis of the numerical solution are made a practical and simple algorithm is given to obtain the approximation solution a numerical example is given to verify the proposed method and compare with the known existing ones

a new algorithm for integral of trigonometric functions with mechanization in this paper by using the theories of mathematical analysis the authors systematically studied the problem of integrating of trigonometric functions utilizing omnipotence expressions tan x2 t a new integral algorithm for the integral of trigonometric functions was established and a new maple program trigint for the calculation mechanizing for the integral of trigonometric functions was established trigint could evaluate not only the integral of trigonometric functions utilizing omnipotence expresssions correctly but also all the other kinds of functions the results indicated that the algorithm established in this paper had advantages of simple idea excellent property for operation and powerful competence this would be useful for some calculating problems of engineering mathematics

convergence and extrapolation in finite size scaling renormalization abstract finite size renormalization technique is studied by examining a test model which is exactly solvable and the problem of lattice animals and self avoiding walks bulk estimates of the leading irrelevant rg eigenvalue are found and compared with independent estimates from finite size data we introduce an extrapolation technique which is applicable to non monotonic sequences of exponent approximants

multiple phase coexistence and the scaling transformation it is shown that a simple extension of the finite size scaling method in the theory of critical phenomena can yield a sequence of approximants to any point subset of the interaction parameter space where multiple phase coexistence is possible the method is illustrated by an application to the three four and five state potts models and even in its lowest order of approximation is able to distinguish the difference in the zero field critical behaviour between the five state and the three and four state models

finite size studies of the ashkin teller model the authors study the ashkin teller model in the time continuous hamiltonian version finite size scaling is used to calculate the magnetic gamma m electric gamma p and thermal alpha and nu critical exponents for several values of the coupling constant lambda the results confirm the believed extended scaling relations and suggest a conjecture relating the mass gap amplitudes and critical indices in the hamiltonian context

latent heats from finite size scaling a method of obtaining the latent heat at a first order transition from finite lattice matrix elements is discussed and demonstrated for the case of the 1 1 d potts model the method is shown to provide smoothly convergent finite lattice sequences the structure of the finite lattice hamiltonian eigenvalues is exhibited and a characteristic signal of the first order transition is proposed the finite size scaling behaviour at the transition is also discussed

on the numerical approximation of phase portraits near stationary points we show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one step methods or multi step methods satisfying a strong root condition this means that any continuous trajectory can be approximated by an appropriate discrete trajectory and vice versa to the correct order of convergence and uniformly on arbitrarily large time intervals in particular the stable and unstable manifolds of the discretization converge to their continuous counterparts

convergence of finite size scaling renormalisation techniques abstract the rate of convergence of the transfer matrix finite size scaling or phenomenological renormalisation method is studied it is shown both heuristically and numerically that the convergence of estimates for exponents etc is governed asymptotically by the leading irrelevant variable scaling exponent the more rapid apparent convergence rates observed in many practical calculations for two dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms

computations of multi resultant with mechanization abstract resultant is a classical algebraic tool for determining whether or not a polynomial systems have a common root without explicitly solving for the roots but the complexity of resultants can be very high in practice in this paper by using the theories and methods of computer algebra two reliable algorithms for computing multi resultant problems dixon resultant and macaulay resultant were established two new maple procedures dixonres and macaulayres were established too some examples are presented to illustrate the implementations of the programs this would be useful for solving a system of polynomial equations

numerical solution of linear fredholm integral equation by using hybrid taylor and block pulse functions in this paper we use a combination of taylor and block pulse functions on the interval 0 1 that is called hybrid functions to estimate the solution of a linear fredholm integral equation of the second kind we convert the integral equation to a system of linear equations and by using numerical examples we show our estimation have a good degree of accuracy

semi hyperbolic mappings condensing operators and neutral delay equations abstract semi hyperbolic mappings in banach spaces are lipschitz continuous and not necessarily invertible like hyperbolic mappings they involve a splitting into stable and unstable spaces but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional it is shown that semi hyperbolic mappings are locally contracting where is the hausdorff measure of noncompactness and that a linear operator is semi hyperbolic if and only if it is contracting and has no spectral values on the unit circle a bishadowing result which combines both direct and indirect forms of shadowing is extended to semi hyperbolic mappings in banach spaces with locally condensing continuous comparison mappings the result is applied to linear neutral delay equations with nonsmooth perturbations

on the hartman grobman theorem for maps it is the purpose of this paper to extend this result to the case where l is a hyperbolic endomorphism i e not necessarily bijective this situation usually occurs in applications to partial differential equations before stating our results we introduce some notation if x is a banach manifold then x denotes the banach manifold of all functions from e to x any map f x x induces a map 3 x x through as f 0 s for sex we set a f n f cxl n o

a critical amplitude relation for one dimensional quantum transitions and determination of the exponent eta a new critical amplitude relation which holds for one dimensional quantum ground state transitions is presented the relation yields an estimate of the sound velocity which appears in the conformal field theory as an a priori unknown parameter with the use of this relation the exponent eta can be explicitly determined from only the scaling fit of the off critical energy gap the relation is confirmed in the transverse ising model the s 1 2 anisotropic xy model and the three state potts model

conformal invariance and the operator content of the xxz model with arbitrary spin the authors are concerned with the critical properties of the anisotropic heisenberg chain or xxz model with arbitrary integer or half integer spin the eigenspectra of these hamiltonians with periodic boundaries are calculated for finite chains by solving numerically their associated bethe ansatz equations the resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified for lattices with an even and odd number of sites the results for spin 1 and spin 3 2 indicate that the conformal anomaly c for these models in the gapless regime has the value c 3s 1 s independent of the anisotropy and the exponents vary continuously with the anisotropy as in the eight vertex model the operator content of these models indicate that the underlying field theory governing these critical spin s models is described by composite fields formed by the product of gaussian and z n fields with n 2s finally some of the irrelevant operators which produce the leading finite size corrections of the eigenenergies are identified

linearization and invariant manifolds on the carrying simplex for competitive maps abstract a result due to m w hirsch states that most competitive maps admit a carrying simplex i e an invariant hypersurface of codimension one which attracts all nontrivial orbits the common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex however this manifold is normally non smooth therefore not every tool coming from differential geometry can be applied in this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo unstable manifold by an invariant foliation this implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex we further construct the stable and unstable manifolds on the carrying simplex our results give partial responses to hirsch s problem regarding the smoothness of the carrying simplex we discuss some applications in classical models of population dynamics

a family of methods for the solution of lattice models abstract we discuss a family of methods for evaluating the thermodynamic functions for a class of lattice models that includes the ising model it is shown that the usual transfer matrix method can be considered as a member of the family some typical members of this family are also introduced furthermore we are concerned with the acceleration of these methods for this reason an extra linear extrapolation algorithm which is based on the member of this family related to the transfer matrix method is presented by this algorithm not only can the bulk physical quantities be evaluated more precisely but also an approximation to the boundary thermodynamic quantities can be obtained a new expression for the boundary free energy of the ising model is derived in addition the convergence to the thermodynamic limit is proven for the scaling method presented by chorin some numerical results for the bulk and boundary thermodynamic functions of the ising model are included

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach