continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

finite size scaling analysis of the dilute baxter wu model the finite size scaling method is used to study the critical properties of the spin 1 baxter wu model as function of the fugacity z of the vacant spin zero sites for z 0 the thermal exponent converges very quickly to the exact value yt 3 2 for z 0 yt monotonically increases beyond the value 2 this increase is interpreted as indicating a first order transition out of several possible renormalisation group flows the results seem to favour the one in which the critical baxter wu hamiltonian flows to the fixed point of the 4 state potts model with the amplitude of the marginal operator equal to zero

renormalisation group approach to delaunay percolation networks with topological disorder a renormalisation group approach is developed for delaunay percolating systems in two and three dimensions using a scaling transformation for a finite lattice in real space considering various renormalisation transformations for two and three dimensional delaunay lattices the authors determine the behaviour of the probabilities under a scale transformation and calculate the fixed point and connectedness length exponent the fixed points for the two dimensional bond lattice and the three dimensional site lattice are 0 3229 and 0 1443 respectively which are in excellent agreement with results of monte carlo simulations the fixed point for the two dimensional site lattice gives the value 1 2 for the critical percolation probability which is equal to the known result for a fully triangulated lattice

correlated percolation and a sublattice dilute potts model at t 0 abstract it is shown that a sublattice dilute q state potts model at t 0 is equivalent to a correlated percolation which favours subgraphs with larger numbers of disconnected finite clusters the syozi model at t is equivalent to the above model with q 2 in the same limit and thus corresponds to the same correlated percolation a table is listed which reflects the influence of correlation on critical probabilities for such correlated percolation processes on some lattices

position space renormalisation group for isolated polymer chains we develop a cell position space renormalisation group psrg with which we study the scaling properties of isolated polymer chains we model a chain by a self avoiding walk constrained to a lattice for rescaling factors b s 6 we calculate recursion relations analytically on the square lattice with several different choices for the psrg weight function we also calculate implicit cell to cell transformations in which a cell of size b is rescaled to a cell of size b the results of these psrgs improve both as b increases and as b b 1 in addition we construct a true infinitesimal psrg transformation which appears to become exact as the dimensionality d approaches 1 we obtain the closed form expression for the correlation length exponent u d l d in d the flory formula deviates from this already at first order in d 1 we also develop a constant fugacity monte carlo method which enables us to simu late in an unbiased way within the grand canonical ensemble chains of up to lo3 bonds with this method we extend the psrg to larger cells b s 150 on the square lattice our numerical method provides high statistical accuracy for all cell sizes however in the range of b we study the asymptotic behaviour of our results appears to depend on the choice of weight function one weight function provides smooth behaviour as a function of b and with it we extrapolate to find u 0 756 0 004 further work is required to resolve the apparent anomalies in the results based on the other weight functions

fast geodesic active fields for image registration based on splitting and augmented lagrangian approaches in this paper we present an efficient numerical scheme for the recently introduced geodesic active fields gaf framework for geometric image registration this framework considers the registration task as a weighted minimal surface problem hence the data term and the regularization term are combined through multiplication in a single parametrization invariant and geometric cost functional the multiplicative coupling provides an intrinsic spatially varying and data dependent tuning of the regularization strength and the parametrization invariance allows working with images of nonflat geometry generally defined on any smoothly parametrizable manifold the resulting energy minimizing flow however has poor numerical properties here we provide an efficient numerical scheme that uses a splitting approach data and regularity terms are optimized over two distinct deformation fields that are constrained to be equal via an augmented lagrangian approach our approach is more flexible than standard gaussian regularization since one can interpolate freely between isotropic gaussian and anisotropic tv like smoothing in this paper we compare the geodesic active fields method with the popular demons method and three more recent state of the art algorithms nl optical flow mrf image registration and landmark enhanced large displacement optical flow thus we can show the advantages of the proposed fastgaf method it compares favorably against demons both in terms of registration speed and quality over the range of example applications it also consistently produces results not far from more dedicated state of the art methods illustrating the flexibility of the proposed framework

finite time scaling via linear driving application to the two dimensional potts model we apply finite time scaling to the q state potts model with q 3 and 4 on two dimensional lattices to determine its critical properties this consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields from the finite time scaling of the order parameter the coercivity and the hysteresis area and its derivative we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature the static critical exponents obtained in general and the magnetic exponent ensuremath delta in particular agree reasonably with the conjectured ones the dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short time dynamic results for q 4 our results also suggest an alternative way to characterize the weak universality

efficient estimation of first passage time density function for jump diffusion processes the first passage time problem has attracted considerable research interest in the field of stochastic processes it concerns the estimation of the probability density of the time for a random process to cross a specified boundary level even though there are many theoretical advances in solving this problem for many classes of random processes no analytical solution exists the jump diffusion process jdp is one such class recent research in finance theory has renewed the interest in jdps and the first passage time problem for such processes is applicable to several finance problems in this paper we develop fast monte carlo type numerical methods for computing the first passage density function for jdps compared with the standard monte carlo approach our approaches are about 10 30 times faster

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

a taylor galerkin method for convective transport problems a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions to produce accurate temporal differencing the method employs forward time taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation this yields a generalized time discretized equation which is successively discretized in space by means of the standard bubnov galerkin finite element method the technique is illustrated first in one space dimension with linear elements and euler leap frog and crank nicolson time stepping several interesting relations with standard galerkin and recently developed petrov galerkin methods emerge and the new taylor galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping the method is successively extended to deal with variable coefficient problems and multi dimensional situations

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

the piecewise exponential method pem for the numerical simulation of wave propagation a multidimensional wave problem abstract this paper presents the extension of the pem scheme for the case of multidimensional wave problems this is done on the basis of operator splitting and the pem scheme for the one dimensional wave equation given in the previous publication v zalizniak commun numer meth engrg 13 3 1997 163 171 numerical results for two problems of wave propagation in comparison with the analytical solutions are given to demonstrate the effectiveness of the proposed method for elastodynamics

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

phase equilibria and the scaling transformation it is argued that the scaling transformation which is currently being used to obtain numerical estimates of critical exponents cannot safely be employed without prior knowledge that the points in question are second order transition points the transformation when defined on finite or semi infinite systems will not distinguish between order disorder critical points and any other point on the coexistence surface this in turn means that the full phase diagram of the equilibrium surface can be found using the scaling transformation the three state potts model is used to illustrate this

stability and convergence of collocation schemes for retarded potential integral equations time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations solving such equations numerically is both complicated and computationally intensive and numerical methods often prove to be unstable collocation schemes are easier to implement than full finite element formulations but little appears to be known about their stability and convergence here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering and use spatial fourier and temporal laplace transform techniques to demonstrate that such stable schemes are second order convergent

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

ising like critical phenomena in two dimensional percolation numerical evidence a recent analytic theory of two dimensional isotropic percolation indicates that the singular behaviour near pc is identical to that appropriate for the associated dilute ising critical point a specific prediction supported by recent series expansion studies is that the mean number of clusters k p presents a singularity of the form k p approximately mod p pc mod 2ln mod ln mod p pc rather than the accepted form k p approximately mod p pc mod 2 alpha with alpha 2 3 a novel numerical and renormalisation group finite size scaling analysis of the nature of the singularity in k p is presented in support of the new theory which implies the absence of a separate universality class for two dimensional percolation processes this study is consistent with the effective values of the exponents alpha 0 and nu 1

critical behaviour and conformal anomaly of the o n model on the square lattice finite size scaling and transfer matrix techniques are used to determine the conformal anomaly and critical exponents of o n models on the square lattice these calculations were performed on five branches of critical points parametrised by n the results for two of the branches agree with the known universal properties of the o n model as derived for the honeycomb lattice finite size convergence is poor for most of the third branch so that in this case the classification in terms of known models is uncertain the authors data show that the fourth branch can be interpreted in terms of a combination of a low temperature o n and a critical ising model finally the numerical results for the o n model on the remaining branch agree accurately with the known results for the low temperature o n 1 model they prove this equivalence by means of an exact mapping

jacobi iteration in implicit difference schemes for the wave equation it is shown how a jacobi iteration can be used to approximate the linear algebra required by a class of implicit finite difference element schemes for a linear hyperbolic pde while preserving the important properties of the implicit scheme in particular it is shown that order of accuracy stability energy conservation and isotropy of the implicit scheme are all carried over into the new scheme in general the new scheme is much more efficient than the implicit scheme it is derived from for problems in more than one space variable and can be vectorized and or parallelized quite easily furthermore it is shown that a modification of the technique can be used to improve some properties of the underlying implicit scheme at little or no extra cost

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

time accurate solution of advection diffusion problems by finite elements abstract the taylor galerkin method recently proposed for the spatial and temporal discretization of hyperbolic equations is employed to derive accurate and efficient numerical schemes for the solution of time dependent advection diffusion problems two distinct numerical strategies are discussed the first is suitable for evolutionary problems while the second is appropriate for situations in which a steady state is eventually reached for purely evolutionary problems the taylor galerkin method is applied to the complete advection diffusion equation and through modifications of the standard mass matrix is shown to generate an incremental form of the crank nicolson time stepping method such an incremental form also lends itself to the derivation of computationally simple but accurate explicit algorithms to deal with transient situations which evolve towards a highly convective steady state the global taylor galerkin method is found to be ineffective since it reduces to the standard galerkin formulation as the temporal term vanishes for such cases we suggest the use of a splitting up method in which advection and diffusion are treated separately by appropriate taylor galerkin methods the proposed methods are illustrated first for linear constant coefficient equations they are successively extended to deal with nonlinear and multi dimensional problems numerical applications indicate their effectiveness

anisotropy propagation failure and wave speedup in traveling waves of discretizations of a nagumo pde this article is concerned with effect of spatial and temporal discretizations on traveling wave solutions to parabolic pdes nagumo type possessing piecewise linear bistable nonlinearities solution behavior is compared in terms of waveforms and in terms of the so called a c relationship where a is a parameter controlling the bistable nonlinearity by varying the potential energy difference of the two phases and c is the wave speed of the traveling wave uniform spatial discretizations and a stable linear multistep methods in time are considered results obtained show that although the traveling wave solutions to parabolic pdes are stationary for only one value of the parameter a a0 spatial discretization of these pdes produce traveling waves which are stationary for a nontrivial interval of a values which include a0 i e failure of the solution to propagate in the presence of a driving force this is true no matter how wide the interface is with respect to the discretization for temporal discretizations at large wave speeds the set of parameter a values for which there are traveling wave solutions is constrained an analysis of a complete discretization points out the potential for nonuniqueness in the a c relationship

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

monte carlo method for critical systems in infinite volume the planar ising model in this paper we propose a monte carlo method for generating finite domain marginals of critical distributions of statistical models in infinite volume the algorithm corrects the problem of the long range effects of boundaries associated to generating critical distributions on finite lattices it uses the advantage of scale invariance combined with ideas of the renormalization group in order to construct a type of holographic boundary condition that encodes the presence of an infinite volume beyond it we check the quality of the distribution obtained in the case of the planar ising model by comparing various observables with their infinite plane prediction we accurately reproduce planar two three and four point of spin and energy operators we also define a lattice stress energy tensor and numerically obtain the associated conformal ward identities and the ising central charge

anisotropic cubic lattice potts ferromagnet renormalisation group treatment the choice of a convenient self dual cell within a real space renormalisation group framework enables a satisfactory treatment of the anisotropic square lattice q state potts ferromagnet criticality the exact critical frontier and dimensionality crossover exponent hf as well as the expected universality behaviour renormalisation flow sense are recovered for any linear scaling factor b and all values of q q or 4 the b 2 and b 3 approximate correlation length critical exponent nu is calculated for all values of q and compared with the den nijs conjecture the same calculation is performed for all values of b for the exponent nu d 1 associated with the one dimensional limit and the exact result nu d 1 1 is recovered in the limit b to infinity