monte carlo generation of self avoiding walks with fixed endpoints and fixed length we propose a new class of dynamic monte carlo algorithms for generating self avoiding walks uniformly from the ensemble with fixed endpoints and fixed length in any dimension and prove that these algorithms are ergodic in all cases we also prove the ergodicity of a variant of the pivot algorithm

multi stage solvers optimized for damping and propagation explicit multi stage solvers are routinely used to solve the semi discretized equations that arise in computational fluid dynamics cfd problems often they are used in combination with multi grid methods in that case the role of the multi stage solver is to efficiently reduce the high frequency modes on the current grid and is called a smoother in the past when optimizing the coefficients of the scheme only the damping characteristics of the smoother were taken into account and the interaction with the remainder of the multi grid cycle was neglected recently it had been found that coefficients that result in less damping but allow for a higher courant friedrichs lewy cfl number are often superior to schemes that try to optimize damping alone while this is certainly true for multi stage schemes used as a stand alone solver we investigate in this paper if using higher cfl numbers also yields better results in a multi grid setting we compare the results with a previous study we conducted and where a more accurate model of the multi grid cycle was used to optimize the various parameters of the solver we show that the use of the more accurate model results in better coefficients and that in a multi grid setting propagation is of little importance we also look into the gains to be made when we allow the parameters to be different for the pre and post smoother and show that even better coefficients can be found in this way

multi staging of jacobi relaxation in multigrid methods for steady euler equations ii multi stage versions of jacobi relaxation are studied for use in multigrid methods for steady euler equations in particular versions adapted to the nonlinear tvd discretization are analysed it is shown that multi stage relaxation versions lead to more efficient multigrid methods than possible with classic relaxation methods

a generalization of the runge kutta iteration iterative solvers in combination with multi grid have been used extensively to solve large algebraic systems one of the best known is the runge kutta iteration we show that a generally used formulation a jameson numerical solution of the euler equations for compressible inviscid fluids in f angrand a dervieux j a desideri r glowinski eds numerical methods for the euler equations of fluid dynamics siam philadelphia 1985 pp 199 245 does not allow to form all possible polynomial transmittance functions and we propose a new formulation to remedy this without using an excessive number of coefficients after having converted the optimal parameters found in previous studies e g b van leer c h tai k g powell design of optimally smoothing multi stage schemes for the euler equations aiaa paper 89 1923 1989 we compare them with those that we obtain when we optimize for an integrated 2 grid v cycle and show that this results in superior performance using a low number of stages we also propose a variant of our new formulation that roughly follows the idea of the martinelli jameson scheme a jameson analysis and design of numerical schemes for gas dynamics 1 artificial diffusion upwind biasing limiter and their effect on multigrid convergence int j comput fluid dyn 4 1995 171 218 j v lassaline optimal multistage relaxation coefficients for multigrid flow solvers http www ryerson ca jvl papers cfd2005 pdf used on the advection diffusion equation which that can be extended to other types gains in the order of 30 50 have been shown with respect to classical iterative schemes on the advection equation better results were also obtained on the advection diffusion equation than with the martinelli jameson coefficients but with less than half the number of matrix vector multiplications

phase diagram of the ising model on a cayley tree in the presence of competing interactions and magnetic field we study the phase diagram for the ising model on a cayley tree with competing nearest neighbor interactionsj1 and next nearest neighbor interactionsj2 andj3 in the presence of an external magnetic field to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular cases previous works by vannimenus and by inawashiroet al at vanishing temperature the phase diagram is fully determined for all values and signs ofj2 j1 andj3 j2 in particular we verify that values ofj3 j2 high enough favor the paramagnetic phase at finite temperatures several interesting features evolution of reentrances separation of the modulated region into two disconnected pieces etc are exhibited for typical values ofj2 j1 andj3 j2

optimization of the runge kutta iteration with residual smoothing iterative solvers in combination with multi grid have been used extensively to solve large algebraic systems one of the best known is the runge kutta iteration previously haelterman et al 2009 3 we reformulated the runge kutta scheme and established a model of a complete v cycle which was used to optimize the coefficients of the multi stage scheme and resulted in a better overall performance we now look into aspects of central and upwind residual smoothing within the same optimization framework we consider explicit and implicit residual smoothing and either apply it within the runge kutta time steps as a filter for restriction or as a preconditioner for the discretized equations we also shed a different light on the very high cfl numbers obtained by upwind residual smoothing and point out that damping the high frequencies by residual smoothing is not necessarily a good idea

study of the shape of random walks a comprehensive study of the shape of discrete random walks considering the general case of arbitrary length and any space dimension is presented the probability distributions for several magnitudes such as the principal inertia moments the asphericity and the angle between the principal axis of inertia and the end to end vector are evaluated numerically in most cases and especially in low dimensional spaces these probability distributions spread widely and possess a non zero skewness this implies that any description of the shapes of random walks which is based only on mean values of related magnitudes is incomplete this situation does nor vary when random walks of large lengths are considered since the relative fluctuations ratio between the standard deviation and the mean value do not go to zero in that limit on the other hand when the space dimension grows the distributions become more peaked reaching the expected asymptotic limit for infinite dimension the present study also indicates that the probability distributions for the principal inertia moments have approximately the form of chi squared distributions analysing the probability distribution for the asphericity it is shown that the distributions for the different moments are not completely independent

the shapes and sizes of two dimensional pressurized self intersecting rings as models for two dimensional vesicles two dimensional vesicles are modelled as pressurized unrestricted closed random walks the model though suffering from undesirable physical characteristics has the decided advantage of yielding exact analytical statistical analysis of the size and shapes of these objects closed form expressions for the average of the square of the radius of gyration and the asphericity are derived in addition exact expressions for the full probability distribution are also obtained numerical computations of the probability function for several pressures are presented the authors arrive at a new exact formula for the distribution function p r12 r22 of the principal components of the square of the radius of gyration for two dimensional rings

the shapes of open and closed random walks a 1 d expansion a new technique for calculating the shapes of random walks is presented the method is used to derive an exact analytical expression for the asphericity of an unrestricted closed or ring walk embedded in d spatial dimensions a graphical procedure is developed to systematise a 1 d series expansion for the individual principal radii of gyration and their respective probability distribution functions p r2i 1 or i or d the average principal radii of gyration are calculated to o 1 d2 for both open and closed walks and selected terms in the 1 d expansion are summed to all orders in 1 d in the determination of p r2i this leads to an explicit analytical form for p r2i for open walks the distribution of the largest eigenvalue is compared with a distribution obtained from numerical simulations of walks in three dimensions the agreement between the two is extremely good other predictions for various parameters that characterise the average shape of open and closed walks in three dimensions are also found to agree remarkably well with the results of simulations the error being of the order of 5

the shapes of high dimensional random walks the individual principal radii of gyration are computed for unrestricted random walks in a large number of spatial dimensions a low order expansion in one over the dimensionality yields useful information regarding the distribution and average of these measures of the extent and spatial anisotropy of high dimensional random walks such an expansion may well prove useful in the study of the shapes of other random fractal objects

optimizing a multigrid runge kutta smoother for variable coefficient convection diffusion equations abstract the theory of generally locally toeplitz or glt for short sequences of matrices is proposed in the analysis of a multigrid solver for the linear systems generated by finite volume finite difference approximations of variable coefficients linear convection diffusion equations in 1d proposed by birken in 2012 and extended here to 2d problems the multigrid solver is used with a runge kutta smoother optimal coefficients for the smoother are found by considering the unsteady linear advection equation and using optimization algorithms in particular in order to reduce the issues of having multiple local minima the sequential quadratic programming sqp mixed with genetic and particle swarm optimization algorithms are proposed numerical results show that our proposals are competitive with respect to other multigrid implementations

solution of the euler equations for two dimensional transonic flow by a multigrid method the assumption of potential o w implies that the o w is irrotational this is not strictly correct when shock waves are present an exact description of transonic inviscid o w requires the solution of the euler equations the numerical solution of the euler equations for steady transonic o ws is therefore a problem of great interest to the aeronautical community it also presents a testing challenge to applied mathematicians and numerical analysts

competing binary and k tuple interactions on a cayley tree of arbitrary order abstract we study the phase diagram for the ising model on a cayley tree of arbitrary order k with competing nearest neighbor interactions j 1 prolonged next nearest neighbor interactions j p and one level k tuple neighbor interaction j o the phase diagram is studied for several ranges of the competing parameters it shows the appearance of several features and modulated phases arising from the frustration effects introduced by the one level k tuple neighbor interaction j o the variation of the wavevector with temperature in the modulate phase is studied in detail it shows narrow commensurate steps between incommensurate regions finally the lyapunov exponent associated with the trajectory of the system is investigated

study of the shape of random walks ii inertia moment ratios and the two dimensional asphericity a study of some features of the quantities used to describe the shape of random walks focusing on the probability distribution of the asphericity of discrete random walks in two dimensions is presented a connection is established between the asphericity distribution and the probability distribution corresponding to the ratio of the principal inertia moments the probability distributions of arbitrary inertia moment ratios are analysed for varying spatial dimension and an analytic expression for them is found which presents excellent agreement with monte carlo data in all the cases considered this function is then used to obtain the corresponding two dimensional asphericity distribution which also proved to be a very good approximation to the true distribution obtained from an independent monte carlo simulation

local properties of an ising model on a cayley tree the ising model on a cayley tree displays a peculiar continuous order phase transition with zero long range order at all finite temperatures when one studies expection values of spins far removed from the surface which contains a finite fraction of the total number of spins in the thermodynamic limit however one obtains the so called bethe approximation here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface in the ferromagnetic case the local magnetization is a fixed point of the iterative transformation while in the antiferromagnetic case the fixed point bifurcates to a two cycle of the transformation for low temperatures and fields giving rise to local sublattice magnetizations in both cases local thermodynamical properties are obtained by integration

optimal runge kutta smoothers for the p multigrid discontinuous galerkin solution of the 1d euler equations this work presents a family of original runge kutta methods specifically designed to be effective relaxation schemes in the numerical solution of the steady state solution of purely advective problems with a high order accurate discontinuous galerkin space discretization and a p multigrid solution algorithm the design criterion for the construction of the runge kutta methods here developed is different form the one traditionally used to derive optimal runge kutta smoothers for the h multigrid algorithm which are designed in order to provide a uniform damping of the error modes in the high frequency range only the method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum the performance of the proposed schemes is assessed in the solution of the steady state quasi one dimensional euler equations for two test cases of increasing difficulty some preliminary results showing the performance for multidimensional applications are also presented

the shape of self avoiding walks an exhaustive analysis of the shape properties of discrete self avoiding random walks is presented the dependence of the main parameters of the probability distributions from the length of the walk and the differences between these distributions for conventional and self avoiding walks are studied by means of high precision monte carlo simulations it is shown that the characteristics of the shape of self avoiding random walks when regarded as a function of the walk length present the expected asymptotic behaviour the differences with conventional random walks depend upon the observable considered the probability distributions of the principal inertia eigenvalues of self avoiding walks spread around the most probable value more widely than the corresponding distributions for unrestricted walks while in the cases of the asphericity or ratios of inertia eigenvalues the distributions for self avoiding walks are somewhat more peaked than their counterparts for conventional random walks analytical expressions for the probability distributions of inertia moment ratios and the two dimensional asphericity are given for common random walks there is an excellent agreement between these analytical distributions and the monte carlo data in this work it is established that this concordance is maintained if the same analytical distributions are applied to the self avoiding case this means that within the precision of the simulations the functional form of the mentioned distributions does not vary when passing from conventional to self avoiding walks

optimal multistage schemes for euler equations with residual smoothing a numerical technique composed of the van leer tai powell optimization and a modified procedure is applied to design multistage schemes that give optimal damping of high frequencies for given upwind biased spatial differencing with implicit and explicit residual smoothing the analysis is done for a scalar convection equation in one space dimension the object of this technique is to make the schemes suited for multigrid acceleration the optimal multistage schemes and their damping properties are presented in this paper by keeping the multistage coefficients from the one dimensional analysis and simply redefining the courant number the schemes can be applied to multidimensional problems a fast euler code is used to evaluate the suitability of the schemes for multidimensional multigrid computations numerical results show that the modified schemes effectively enhance the convergence performance on both single and multiple grids

on uniqueness of gibbs measure for p adic countable state potts model on the cayley tree abstract in the present paper we consider countable state of p adic potts model on the tree under some condition on weights we establish uniqueness of gibbs measures for the model note that this condition does not depend on values of the prime p an analogous fact is not true when the number of spins is finite

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