diffusion on self avoiding walk networks two properties of self avoiding walk networks constructed by connecting the nearest neighbour points of self avoiding walks on the square lattice are studied numerically the average end to end linear part of such networks is investigated and is found to be proportional to the length of the walk secondly random walks on this structure are studied and the associated spectral dimension of such networks is estimated to be 1 03 or 0 03 which implies that the effect of such connections affects the spectral dimension only weakly if at all

remarks on the possible universal mechanism of the non linear long term autocorrelations in financial time series the paper consists of two parts i the empirical one where the non linear long term autocorrelations present in high frequency data extracting from the warsaw stock exchange were analyzed and ii theoretical one where predictions of our model quantitative finance 3 2003 201 physica a 2003 chem phys 284 2002 481 phys comm 147 2002 565 physica a 264 1999 84 physica a 264 1999 107 lecture notes in computer science 2657 2003 407 eur phys j b 33 2003 495 were shown and discussed this model introduces the possibility that the weierstrass hierarchical random walk can be occasionally intermitted by momentary localizations the localizations themselves are again described by the weierstrass process in other words this combined walk is a kind of the non separable generalized continuous time random walk formalism to adapt the model to the description of empirical data recorded at time horizon t 1min we applied a discretization procedure into the continuous time series produced by the model we observed that such a procedure generates the non linear long term autocorrelations even in the gaussian regime as turning points of the random walk trajectory are most often incommensurable with discretization time step these autocorrelations appear to be similar to those observed in the financial time series physica a 287 2000 396 physica a 299 2001 1 physica a 299 2001 16 physica a 299 2001 28 although single steps of the walker within continuous time are by definition uncorrelated our approach suggests a surprising origin of the non linear long term autocorrelations alternative to the one proposed very recently cf phys rev e 67 2003 021112 and refs therein although both approaches involve related variants of the well known ctrw formalism applied in yet many different branches of knowledge phys rep 158 1987 263 phys rep 195 1990 127 in a bunde s havlin eds fractals in science springer berlin 1995 pp 119 161

to what class of fractals does the alexander orbach conjecture apply discussion de la decouverte numerique recente d alexander et orbach selon laquelle pour un agregat infini naissant en percolation le rapport d f d w dimension fractale de l agregat dimension fractale d une marche aleatoire dans l agregat est approximativement independant de d pour d 1

correction to scaling exponents and amplitudes for the correlation length of linear polymers in two dimensions we consider the scaling behaviour of the radius of gyration for a system of dilute linear polymers we focus on pn the mean square end to end distance of an n step self avoiding walk for which an additional two terms were recently calculated for the close packed triangular lattice combining several extrapolation methods we find that a consistent description of the scaling behaviour exists if and only if the correction to scaling exponent a is roughly half as large as commonly believed we conclude that all data ale consistent with the equation pn an2 1 b na c n where y a a 1 j2 abso 21 and c lo

monte carlo study of random walks on a 2d gasket fractal in an external field using monte carlo simulation random walks on a two dimensional sierpinski gasket in the presence of an external field were studied it was observed that the random walk motion of a particle displayed a crossover from anomalous diffusion to drift for a non zero bias field and the crossover time tcr was a decreasing function of the external bias field the associated dynamic exponents obtained in a computer simulation agree with the predictions of stinchcombe s scaling treatment 1985

self avoiding polygons on the square l and manhattan lattices transfer matrix techniques are used to extend the self avoiding polygon generating function on the square lattice to terms in x46 corresponding to 46 step polygons these techniques are then extended to apply to directed square lattices such as the l and manhattan lattice and the self avoiding polygon generating function to x48 is found for these lattices series analysis confirms that the specific heat exponent alpha 1 2 for the self avoiding walk problem and gives the following estimates for the connective constants mu sq 2 638155 or 0 000025 mu l 1 5657 or 0 0019 and mu man 1 7328 or 0 0005 some evidence for a correction to scaling exponent delta approximately 0 84 is found from square lattice series

parallel adaptive gmres implementations for homotopy methods the success of probability one homotopy methods in solving large scale optimization problems and nonlinear systems of equations on parallel architectures may be significantly enhanced by the accurate parallel solution of large sparse nonsymmetric linear systems iterative solution techniques such as gmres k favor parallel implementations however their straightforward parallelization usually leads to a poor parallel performance because of global communication incurred by processors one variation of gmres k considered here is to adapt the restart value k for any given problem and use householder reflections in the orthogonalization phase coupled with graph based matrix partitioning to achieve high accuracy and reduce the communication overhead this particular gmres implementation is tailored to the uniquely stringent requirements imposed on a linear system solver by probability one homotopy algorithms occasionally unusually high accuracy ability to adapt to problems of widely varying difficulty and parallelism

memory flow structures in lorenz chaos memory flow structures of a coupled three element model here a lorenz chaos have been investigated although in the models of coupled two element systems a bidirectional connectivity of information transference is held and the memory flows of one channel exhibit a nearly out of phase relation it is found that in this lorenz chaos one additional unidirectional connectivity can also occur and the nearly out of phase relation in the coupled two element models will be replaced by an antiphaselike relation in lorenz chaos

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

resistance and related scaling properties of self avoiding random walks three scaling exponents for random and self avoiding chains are studied by direct renormalisation these describe the end to end resistance of a conducting chain the arc length of the shortest path traceable between the chain ends and the arc length of chain in no loop the scaling behaviours of these quantities are identical to first order in epsilon 4 d d is the space dimension for a long self avoiding chain the authors predict that each one one is proportional to the chain length

new exact exponents for two dimensional self avoiding walks the author establishes using standard coulomb gas methods the values of a new set of geometrical exponents that have been previously conjectured these exponents give for any p the number of configurations of p two dimensional self avoiding walks of the same length l which are attached by their ends

possible origin of the non linear long term autocorrelations within the gaussian regime abstract in this work we extend the recently considered toy model of weierstrass or levy walks with varying velocity of the walker quantitative finance 3 2003 201 chem phys 284 2002 481 comp phys comm 147 2002 565 phys a 264 1999 84 phys a 264 1999 107 by introducing a more realistic possibility that the walk can be occasionally intermitted by its momentary localization the localizations themselves are again described by the weierstrass or levy process the direct empirical motivation for developing this combined model is for example the dynamics of financial high frequency time series or hydrological and even meteorological ones where variations of the index are randomly intermitted by flat intervals of different length exhibiting no changes in the activity of the system this combined weierstrass walks was developed in the framework of the non separable generalized continuous time random walk formalism developed very recently lecture notes comput sci 2657 2003 407 eur phys j b 33 2003 495 this non markovian two state walking localization model makes possible to cover by the unified treatment a broad band of known up to now types of non biased diffusion from the dispersive one over the normal enhanced ballistic and hyperdiffusion up to the richardson law of diffusion which defines here a part of the borderline which separates the latter from the levy ocean where the total mean square displacement of the walker diverges we observed that anomalous diffusion is characterized here by three fractional exponents one temporal characterizing the localized state and two temporal and spatial characterizing the walking one by considering successive dynamic even exponents we constructed a series of different diffusion phase diagrams on the plane defined by the temporal and spatial partial fractional dynamic exponents characterizing the walking state to adapt the model to the description of empirical data the discrete time series which are collected with a discrete time step we used in the continuous time series produced by the model a discretization procedure we observed that such a procedure generates in general long range non linear autocorrelations even in the gaussian regime which appear to be similar to those observed e g in the financial time series phys a 287 2000 396 phys a 299 2001 1 phys a 299 2001 16 phys a 299 2001 16 although single steps of the walker within continuous time are by definition uncorrelated this suggests a suprising origin of long range non linear autocorrelations alternative to the one proposed very recently cf mosaliver et al phys rev e 67 2003 021112 and refs therein although both approaches involve related variants of the well known continuous time random walk formalism applied yet in many different branches of knowledge phys rep 158 1987 263 phys rep 195 1990 127 in a bunde s havlin eds fractals in science springer berlin 1995 p 1

information and entropy in strange attractors a technique for analyzing time series data from experiments is presented that provides estimates of four basic characteristics of a system 1 the measure theoretic entropy 2 the accuracy of the measurements 3 the number of measurements necessary to specify a system state and 4 the best delay time t to use in order to construct phase space portraits by the method of delays these characteristics are obtained by separating the entropy of measurements into a part due to noise and parts due to deterministic effects for the technique to work the noise associated with each measurement must be independent of the noise associated with all other measurements an algorithm for implementing the analysis is presented with three examples

the shortest path of saws with bridges the shortest paths of self avoiding walks saws with bridge lengths b 1 2 3 and 2 are studied by exact series expansions and by monte carlo simulation methods on the square and simple cubic lattices in the series work saw configurations of up to 24 and 14 steps are generated on the square and simple cubic lattices respectively assuming the shortest path sn between two sites which are separated by n steps along the chain has the scaling form sn approximately an bn1 delta it is found that delta approximately 1 and delta approximately 1 2 on the square and simple cubic lattices respectively independent of the bridge length the problem is also investigated by the mellin pade approximation method using monte carlo data on the square lattice the latter method gives results consistent with those of the series expansion study the authors results are not consistent with earlier predictions

random walk on self avoiding walk in external bias diffusion drift and trapping the effect of external bias on random walks rws on self avoiding walks on square lattices in the presence of bridges is studied by monte carlo simulation the qualitative as well as quantitative features of the latter problem are compared and contrasted with those of rws on percolation clusters in the presence of external bias

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

unequal twins probability distributions do not determine everything it is the common lore to assume that knowing the equation for the probability distribution function pdf of a stochastic model as a function of time tells the whole picture defining all other characteristics of the model we show that this is not the case by comparing two exactly solvable models of anomalous diffusion due to geometric constraints the comb model and the random walk on a random walk we show that though the two models have exactly the same pdfs they differ in other respects like their first passage time distributions their autocorrelation functions and their aging properties

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

conjugate gradient type methods for semilinear elliptic problems with symmetry we study block conjugate gradient methods in the context of continuation methods for bifurcation problems by exploiting symmetry in certain semilinear elliptic differential equations we can decompose the problems into small ones and reduce computational cost on the other hand the associated centered difference discretization matrices on the subdomains are nonsymmetric we symmetrize them by using simple similarity transformations and discuss some basic properties concerning the discretization matrices these properties allow the discrete pure mode solution paths branching from a multiple bifurcation point 0 m n of the centered difference analogue of the original problem to be represented by the solution path branching from the first simple bifurcation point 0 1 1 of the counterpart of the reduced problem thus the structure of a multiple bifurcation is preserved in discretization while its treatment is reduced to those for simple bifurcation of problems on subdomains in particular we can adapt the continuation lanczos algorithm proposed in 1 to trace simple solution paths sample numerical results are reported

monte carlo test of a hyperscaling relation for the two dimensional self avoiding walk the authors simulated self avoiding walks on the square lattice with fixed endpoints by means of a dynamic monte carlo algorithm from these data they obtain an evaluation of the effective coordination number mu and the critical exponents alpha and v they can therefore test the hyperscaling relation 2 alpha dv with a careful estimate of systematic and statistical errors

polymer conformations through wiggling a new monte carlo method is proposed which allows for the efficient generation of equilibrium conformations of polymer chains in two and three dimensions the method treats each site monomer as a potential pivot around which a new conformation may be generated by rotating a portion of the chain the method does not suffer from the severe attrition associated with the simple sampling of self avoiding walks and may be extended to treat the interacting polymer chain the authors find in two dimensions that nu 0 748 or 0 005 exact 0 750 and in three dimensions nu 0 595 or 0 005 series expansion and renormalisation group predict nu approximately 0 588 the end end distances calculated for shorter chains are in good agreement with the exact values from enumeration techniques

on the mean radius of gyration of a polymer chain calculates the intrachain bead to bead square distance of a real polymer chain close to the critical dimensionality d 4 from this the authors obtain the mean square radius of gyration s2 of the polymer as a function of the molecular weight n of the chain and the excluded volume parameter mu a proportionality relationship between s2 and the mean square end to end distance r2 of the coil previously suggested by enumerations of self avoiding walks on lattice is shown to be true an estimate of the universal ratio s2 r2 is given

critical dynamics of heisenberg spins on self avoiding walk chains the dynamical exponent z for the critical spin wave dynamics of nearest neighbour interacting heisenberg spins on a self avoiding walk saw chain is estimated here using a scaling picture and also applying a real space renormalisation group technique to some quasilinear fractal lattices the results indicate z 2dt where d is the fractal dimensionality of the saw chain lattice and t is the exponent for the length of the shortest nearest neighbour connecting path of the saw

on the localization of random walks and vibrational excitations in linear fractal structures abstract we study analytically and numerically the mean probability density n of random walks and the mean vibrational amplitudes r w n of localized excitations on linear random fractals averaged over n configurations we find that for large distances r both functions are characterized by crossovers rc and r c respectively which increase logarithmically with n for r below the respective crossover points in n apru and n a r do not depend on n and are characterized by the exponents u dw dw 1 and 1 while above the crossover the coefficients ap and a decrease logarithmically with n and the exponents become u dmindw dw dmin and dmin where dw is the fractal dimension of the random walk and dmin is the fractal dimension of the shortest path on the fractal structure we discuss the applicability of these results to general random fractal structures

biased random walk on a biased random walk abstract we consider the random walk of a particle along topologically linear channels under the influence of a uniform drift force the channels are generated by the usual biased random walk procedure the resulting mean and mean square displacements of a particle are discussed

conductivity exponent and backbone dimension in 2 d percolation abstract we present high statistics simulations for 2 d percolation clusters in the bus bar geometry at the critical point for site and for bond percolation we measured their backbone sizes and electrical conductivities for all sets of measurements we find large corrections to scaling most of which do not seem to be described by single powers using single power terms for the corrections to scaling of the backbone masses we would obtain fractal dimensions which are different for site and bond percolation while the correct result is d b 1 6432 0 0008 for both for the conductivity the corrections to scaling are strongly non monotonic for bond percolation the exponent t t is measured as 0 9826 0 0008 in disagreement with the alexander orbach and other conjectures

resistance of the edwards walk an analytical and numerical study of the scaling of edwards random walks defined as the zero component limit n to 0 of an n component g0 mod phi 2 mod 2 theory is done in three dimensions the hausdorf dimension 1 nu and the resistance exponent x is calculated using renormalisation group methods in 4 in dimensions the numerical study is done by defining a monte carlo algorithm on the cubic lattice to generate ensembles of edwards walks

structure of clusters generated by random walks the authors study the cluster structure resulting from a nearest neighbour random walk embedded in a d dimensional space each bond visited by the random walks is regarded as belonging to the cluster the diffusion exponent and the fracton dimensional of the fractal cluster in d 3 is found to be dw 3 5 or 0 1 and d 0 57 or 0 02 using a method of exact enumeration of random walks on these fractals

the extension of self avoiding random walk series in two dimensions the authors have extended the enumeration of square lattice and triangular lattice self avoiding walks and their end to end distance by two terms they have also calculated 17 terms in the series for the resistance of saws with bridges on the square lattice analysis of the new data allows the connective constants to be more accurately estimated and affirms the values gamma 43 32 and v 3 4 and 1 gamma for the exponent of the resistance series

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

mutual information functions versus correlation functions this paper studies one application of mutual information to symbolic sequences the mutual information functionm d this function is compared with the more frequently used correlation function d an exact relation betweenm d and d is derived for binary sequences for sequences with more than two symbols no such general relation exists in particular d 0 may or may not lead tom d 0 this linear but not general independence between symbols separated by a distance is studied for ternary sequences also included is the estimation of the finite size effect on calculating mutual information finally the concept of symbolic noise is discussed

structural and dynamical properties of random walk clusters the authors study the structural and dynamical properties of the clusters generated by a nearest neighbour random walk embedded in a d dimensional space they have focused on the non trivial case in which the cluster is generated in d 3 the structure of this cluster is characterised by loops for all length scales on the one hand and by the fact that deadends are negligible upon scaling on the other hand the cluster is very dilute and is characterised by fractal dimension df 2 and chemical dimension d1 1 29 or 0 04 from these results it follows that v identical to dldf approximately 2 3 which is consistent with the formula v 2 d 2 or d or 4 obtained using a flory type argument the dynamical diffusion exponents dw and dlw were calculated using the exact enumeration method and found to be dw 3 45 or 0 10 and dlw 2 28 or 0 05 the results suggest that the effect of loops is small but not negligible they also calculated the fracton dimensionality of the cluster and obtained ds 1 14 or 0 02 a scaling function is presented for the end to end mean square displacement of a random walk performed on a random walk cluster this scaling function is supported by their numerical results

extended information in one dimensional maps abstract an information theoretical structure of one dimensional maps is investigated by regarding the maps as a channel rather than a source computing various mutual information numerically for one dimensional dynamical systems we can observe directly the movement of information as to initial conditions one feature of information theoretical structure emerged from this computation is such that initially localized information spreads over whole system as time passes two groups of maps which are contrasting with each other concerning this feature exhibit a striking difference in their transmission ability of information when coupled to form a linear chain

nonergodicity of local length conserving monte carlo algorithms for the self avoiding walk it is proved that every dynamic monte carlo algorithm for the self avoiding walk based on a finite repertoire of local n conserving elementary moves is nonergodic heren is the number of bonds in the walk indeed for largen each ergodic class forms an exponentially small fraction of the whole space this invalidates at least in principle the use of the verdier stockmayer algorithm and its generalizations for high precision monte carlo studies of the self avoiding walk

spectral dimension and the shortest path of saw s with multi neighbour interactions the spectral dimension and the shortest path of self avoiding walks saws with bridge length interaction range b 1 2 3 2 are studied numerically the spectral dimension is calculated by performing exact multi neighbour random walks on the monte carlo generated saw configurations it is found that the spectral dimension is not affected by the finite range of interactions finite bridge length and approach to ds 1 both in dimensions d 2 and 3 the shortest path length sn of the n step saw with local bridges is also investigated it is shown that sn n a n delta b c n the authors numerical simulation results indicate that the exponent delta is independent of the dimensionality and is about 3 16 in dimensions d 2 5

critical properties of the three dimensional self avoiding walk accurate monte carlo data are used to determine the critical exponent nu 0 590 or 0 001 or inverse fractal dimension of the three dimensional self avoiding walk the leading correction to scaling term is non analytic and is near 0 5

shortest path of saws with bridges series results recent monte carlo simulation results of yang and chakrabarti 1990 suggest that the shortest path sn of an n stepped self avoiding walk saw with finite range interactions of bridges has a finite size scaling behaviour sn n approximately a n delta b c n where the exponent delta is superuniversal delta approximately 0 19 for all dimensions d studied 2 or d or 5 the authors report the small n series enumeration results for sn for saws with nearest neighbour bridges on square up to n 18 triangular up to n 11 and simple cubic up to n 12 lattices the estimated values of delta for different lattices approximately 0 22 or 0 01 for d 2 and approximately 0 26 or 0 01 for d 3 have been compared with the above monte carlo estimate and the indication of a superuniversal behaviour for delta has been discussed

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

a direct electrifying algorithm for backbone identification this paper proposes a new algorithm for identifying backbones in the application of percolation theory this algorithm is based on the current carrying definition of backbone and is carried out on the predetermined spanning cluster it is fairly easy to implement and further parallelize the efficiency is enhanced by the fact that the conductivity of a percolating system can be obtained in the same processing of backbone identification the critical exponents of backbone mass red bonds sites and conductivity obtained by this algorithm are in very good agreement with the existing results

preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation the dc operating point of a circuit may be computed by tracking the zero curve of an associated artificial parameter homotopy homotopy algorithms exist that are globally convergent with probability one for the dc operating point problem these algorithms require computing the one dimensional kernel of the jacobian matrix of the homotopy mapping at each step along the zero curve and hence the solution of a linear system of equations at each step these linear systems are typically large highly sparse non symmetric and indefinite several iterative methods which are applicable to such problems including craig s method gmres k bicgstab qmr kacz and lsqr are applied to a suite of test problems derived from simulations of actual bipolar circuits preconditioning techniques considered include incomplete lu factorization ilu sparse submatrix ilu and ilu allowing restricted fill in bands or blocks timings and convergence statistics are given for each iterative method and preconditioner

structure of backbone perimeters of percolation clusters the author studies perimeters of backbones of percolation clusters at the percolation threshold in two dimensions these perimeters are self avoiding for all lattices they simulate these perimeters by three independent processes measure the average size of them and study their variation with length they find the exponent characterising this variation to be 0 744 or 0 012 it seems likely that these perimeters are in the self avoiding walk universality class

small world phenomena and the statistics of linear polymers a regular lattice in which the sites can have long range connections at a distance l with a probabilty p l l in addition to the short range nearest neighbour connections shows small world behaviour for 0 c in the most appropriate physical example of such a system namely the linear polymer network the exponent is related to the exponents of the corresponding n vector model in the n 0 limit and its value is less than c still the polymer networks do not show small world behaviour here we show that this is due to a small value constraint on the number q of long range connections per monomer in the network in the general q space we obtain a phase boundary separating regions with and without small world behaviour and show that the polymer network falls marginally in the regular lattice region

random walk on a random walk the authors investigate the random walk of a particle on a one dimensional chain which has been constructed by a random walk procedure exact expressions are given for the mean square displacement and the fourth moment after n steps the probability density after n steps is derived in the saddle point approximation for large n these quantities have also been studied by numerical simulation the extension to continuous time has been made where the particle jumps according to a poisson process the exact solution for the self correlation function has been obtained in the fourier and laplace domain the resulting frequency dependent diffusion coefficient and incoherent dynamical structure factor have been discussed the model of random walk on a random walk is applied to self diffusion in the concentrated one dimensional lattice gas where the correct asymptotic behavior is found

experiments with conjugate gradient algorithms for homotopy curvetracking there are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points i e with probability one the essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map hompack is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve and has separate routines for dense and sparse jacobian matrices the hompack algorithms for sparse jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernal of the homotopy jacobian matrix a required linear algebra step for homotopy curve tracking here variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with craig s preconditioned conjugate gradient method used in hompack the test problems used include actual large scale sparse structural mechanics problems

confidence interval estimation using standardized time series observations of a stationary stochastic process can be transformed into a standardized time series this paper presents a lemma giving the asymptotic properties of this standardized series under quite general conditions in particular the conditions are satisfied by stationary discrete event simulations confidence intervals can be constructed using this lemma for illustration we develop two easily computed interval estimators for the process mean when independent replications of the series are available such as in computer simulation experiments these interval estimators may be combined with the classical confidence interval estimator these interval estimators also tend to compensate for simulation initialization bias if the sign of the bias is known in an empirical study using three elementary simulated processes the interval estimators presented here compare favorably with the classical interval estimator in a recent paper goldsman and schruben goldsman d l schruben 1982 asymptotic properties of some confidence interval estimators tech rep 544 school of o r i e cornell university ithaca ny show that the asymptotic properties of the confidence intervals presented in this paper strictly dominate those of classical confidence intervals

end to end distance of linear polymers in two dimensions a reassessment recent exact enumeration studies of the two dimensional self avoiding walk have produced evidence in support of non analytic scaling corrections newly extended series for the triangular and square lattices to n 19 and 25 respectively are examined in the light of recent monte carlo results suggesting the absence of corrections of this type using a modified approach to series analysis it is concluded that the end to end distance can best be understood in terms of analytic corrections alone in view of the lack of theoretical support there is little reason to introduce additional kinds of correction in attempting to explain the data

transport properties on a random comb we study the random walk of a particle in a random comb structure both in the presence of a biasing field and an the field free case we show that the mean field treatment of the quenched disorder can be exactly mapped on to a continuous time random walk ctrw on the backbone of the comb with a definite waiting time density we find an exact expression for this central quantity the green function for the ctrw is then obtained its first and second moments determine the drift and diffusion at all times we show that the drift velocity v vanishes asymptotically for power law and stretched exponential distributions of branch lengths on the comb whatever be the biasing field strength for an exponential branch length distribution v is a nonmonotonic function of the bias increasing initially to a maximum and then decreasing to zero at a critical value in the field free case anomalous diffusion occurs for a range of power law distributions of the branch length the corresponding exponent for the mean square displacement is obtained as is the asymptotic form of the positional probability distribution for the random walk we show that normal diffusion occurs whenever the mean branch length is finite and present a simple formula for the effective diffusion constant these results are extended to regular nonrandom combs as well the physical reason for anomalous drift or diffusion is traced to the properties of the distribution of a first passage time on a finite chain that controls the effective waiting time density of the ctrw

new monte carlo method for the self avoiding walk we introduce a new monte carlo algorithm for the self avoiding walk saw and show that it is particularly efficient in the critical region long chains we also introduce new and more efficient statistical techniques we employ these methods to extract numerical estimates for the critical parameters of the saw on the square lattice we find 2 63820 0 00004 0 00030 1 352 0 006 0 025 v 0 7590 0 0062 0 0042 where the first error bar represents systematic error due to corrections to scaling subjective 95 confidence limits and the second bar represents statistical error classical 95 confidence limits these results are based on saws of average length 166 using 340 hours cpu time on a cdc cyber 170 730 we compare our results to previous work and indicate some directions for future research

the self avoiding walk on the honeycomb lattice the author analyses the chain generating function for the self avoiding walk problem on the honeycomb lattice she uses the exact critical point value of nienhuis and finds exponent values of gamma approximately 1 344 and delta 1 approximately 0 93 for the dominant and first confluent exponents the former value is in agreement with nienhuis exact result second and third confluent exponents delta 2 approximately 1 2 and delta 3 approximately 1 5 are also identified these exponents being consistent with monte carlo results and nienhuis exact irrelevant eigenvalue respectively

fourier acceleration of iterative processes in disordered systems technical details are given on how to use fourier acceleration with iterative processes such as relaxation and conjugate gradient methods these methods are often used to solve large linear systems of equations but become hopelessly slow very rapidly as the size of the set of equations to be solved increases fourier acceleration is a method designed to alleviate these problems and result in a very fast algorithm the method is explained for the jacobi relaxation and conjugate gradient methods and is applied to two models the random resistor network and the random central force network in the first model acceleration works very well in the second little is gained we discuss reasons for this we also include a discussion of stopping criteria

universality of symmetric one dimensional random flights the asymptotic behaviour of all the moments of a class of symmetric one dimensional random flights at the first few orders are calculated different symmetric random flights are considered and it is shown that they simply differ up to a scaling in the number of jumps and that important statistical quantities are universal

multiple bifurcations generated by mode interactions in a reaction diffusion problem abstract we study multiple bifurcations in a system of reaction diffusion equations defined on a unit square with robin boundary conditions first we investigate linear stabilities of the system at the uniform steady state solution then we discuss how multiple bifurcations can be generated by mode interactions of the system and how these multiple bifurcations can be preserved in the associated discrete system a continuation unsymmetric lanczos algorithm is described to trace discrete solution curves numerical experiments on the brusselator equations are reported

random walk on self avoiding walk a model for conductivity of linear polymers random walks on self avoiding walks saws are studied using monte carlo techniques on a square lattice with nearest neighbour hopping along the chain and between saw points which are nearest neighbours on the embedding lattice the average of the square of the end to end distance for random walks of t steps on saws of length n is fitted to the scaling forms rt2 varies as ndelta tk for t or approximately ntheta where theta approximately 2 nu s k nu s being the average end to end distance exponent for saws the observed value of the exponent delta is supported by the authors real space renormalisation group result for the conductivity of saw chains the exponent k has been related to the effective fractal dimension of the saw chain

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

real space renormalisation group approach to random walks on self avoiding walks the t dependence of the average of the end to end distance rt2 of t step random walks on self avoiding walks is studied by small cell real space renormalisation the results are in qualitative agreement with earlier monte carlo data

self avoiding walk model for proteins a real space renormalisation group treatment diffusion on the model for proteins introduced by helman coniglio and tsallis 1984 hct is discussed using real space renormalisation group ideas the authors perform cell to cell transformations in two dimensions for both the hct model and the self avoiding walk the latter system is quasi linear and so they expect trivial diffusive behaviour given by dw 2df 8 3 diffusion on the hct model however is complicated by the possibility of jumps across nearest neighbour hydrogen bond bridges the authors results are not consistent with the prediction that dw 2 for the hct model rather they support the assertion that dwhct is close to but slightly greater than dwsaw

spreading and backbone dimensions of 2d percolation the author presents results of high statistics simulations of the spreading of 2d percolation and of backbones of 2d percolation clusters while the algorithm employed for the spreading is more or less standard for identifying backbones the author uses a new and very efficient recursive algorithm the author uses new critical exponents to test previous conjectures

a new algorithm to extract the backbone in a random resistor network the authors present an efficient algorithm to determine the backbone the current carrying part of a lattice in the random resistor network problem it reduces mainly to the determination of the cluster structure on two lattices its efficiency is due to the use of the dual lattice and therefore is specific to the two dimensional case

a real space renormalisation group study of the ising model on self avoiding walk chains here the authors apply a small cell real space renormalisation group technique to study the critical behaviour of the nearest neighbour interacting ising model on self avoiding walk saw chains in the limit of vanishing interactions between nearest neighbours which are not connected by the walk itself the treatment reproduces exactly the critical behaviour of the linear chain ising model when all the nearest neighbour interactions are allowed the treatment gives a vanishing tc with the small cells chosen for the ising model on saws on the square and triangular lattices the study of the critical behaviour of such models indicates some interesting departures from that of the ising model on linear chains

the diffusion of self avoiding random walk in high dimensions we use the brydges spencer lace expansion to prove that the mean square displacement of at step strictly self avoiding random walk in thed dimensional square lattice is asymptotically of the formdt ast approaches infinity ifd is sufficiently large the diffusion constantd is greater than one

scaling behavior of the contact process in networks with long range connections we present simulation results for the contact process on regular cubic networks that are composed of a one dimensional lattice and a set of long edges with unbounded length networks with different sets of long edges are considered that are characterized by different shortest path dimensions and random walk dimensions we provide numerical evidence that an absorbing phase transition occurs at some finite value of the infection rate and the corresponding dynamical critical exponents depend on the underlying network furthermore the time dependent quantities exhibit log periodic oscillations in agreement with the discrete scale invariance of the networks in case of spreading from an initial active seed the critical exponents are found to depend on the location of the initial seed and break the hyperscaling law of the directed percolation universality class due to the inhomogeneity of the networks however if the cluster spreading quantities are averaged over initial sites the hyperscaling law is restored

two dimensional polymers universality and correction to scaling improved monte carlo methods have been used to study the asymptotic behaviour of self avoiding walks on two dimensional lattices the mean square end to end distance rn2 and radius of gyration sn2 are both found to scale as n2v with exponent values very close to the expected v 3 4 and an amplitude ratio that is universal small n deviations of rn2 from the asymptotic results indicate a correction to scaling exponent delta 1 a value that differs from previous estimates

on the correction to scaling exponent of linear polymers in two dimensions the authors present new results which indicate that the leading correction to scaling exponent in the mean squared end to end distance in two dimensions is the analytic term a potential source of the various correction to scaling terms reported in the literature is pointed out

resistance of random walks unrestricted lattice random walks in which a unit conductor is placed along each bond traversed are considered the mean end to end resistance is studied as a function of the number of steps in the walk and the spatial dimension a critical scaling law is found whose exponent is consistently given by four different calculational schemes

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

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

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

surface growth on percolation networks by a conserved noise restricted solid on solid growth model surface growth by the conserved noise restricted solid on solid model is investigated on diluted lattices i e on percolation networks that are embedded in two spatial dimensions the growth exponent and the roughness exponent are defined respectively by the mean square surface width via w 2 t t 2 and the mean square saturated width via w sat 2 l l 2 where l is the system size these are measured on both an infinite network and a backbone network and the results are compared with power counting predictions obtained using the fractional langevin equation while the monte carlo results on deterministic fractal substrates show excellent agreement with the predictions d h kim and j m kim phys rev e 84 011105 2011 the results on critical percolation networks deviate by 8 12 from these predictions

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

the notion of a strange fractal phenomenon disorder induced walk it is argued that a random system coupled with a walk may cause an unusual phenomenon which shares some of the opposite properties of fractals