discrete physical systems discrete physical g phi u systems are defined in terms of a graph g a set of localised states phi and a potential u the dimensionality of the system is defined and shown to be a global property of a graph g the existence of the thermodynamic limit is investigated and shown to exist if g is dimensional and the potential is short ranged

random walks on fractals higher moments the authors investigate by numerical simulation the higher moments for sn the number of distinct sites visited in an n step random walk on fractal structures the sierpinski gasket the sierpinski web and the percolation clusters of the square planar and the simple cubic lattices they find that these moments scale similarly e g the ratio of the standard deviation to sn the average sn is constant in time

random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

fractals in physics squig clusters diffusions fractal measures and the unicity of fractal dimensionality the three topics discussed in this paper are largely independent part 1 fractal squig clusters are introduced and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality physics on these new geometric shapes should prove tractable as background the author s theories of squig intervals and squig trees are reviewed and restated in more versatile form part 2 the notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality scaling noises are touched upon a common formalism is presented for three forms of anomalous diffusion the ant in the fractal labyrinth fractional brownian motion and levy stable motion the fractal dimensionalities common to diverse shapes generated by diffusion are given in table i as functions of the latent dimensionalities of the support of the motion and of the diffusion itself part 3 it is argued that every fractal point set has a unique fractal dimensionality but it is pointed out that many fractals involve diverse combinations of many fractal point sets such is in particular the case for fractal measures and for fractal graphs often called hierarchical lattices the fractal measures that the author had introduced in the early 1970s are described including new developments

a combined integral equation and finite element method for the evaluation of stress intensity factors abstract a combination of the classical finite element and the singular integral equation method with the help of schwarz s alternating method s a m is presented in this way the flexibility of finite element method is complemented by the undisputed power of the singular integral equation method in describing singular fields the result is the creation of a powerful hybrid method convenient for the efficient solution of problems of stress fields containing any type of singularity the examples considered in the paper indicate the excellent approximation obtained by the method even when using only a few classical elements

spectral dimension of a wire network by a rigorous analysis for diffusion on a wire network the spectral dimension of a fractal is shown to be independent of the local structure it is discussed that diffusion does not occur when the lattice spacing tends to zero on such a fractal that the spectral dimension is less than the hausdorff dimension if a free field on the fractal exhibits a certain long distance behaviour for a sierpinski carpet the spectral dimension is evaluated within bond moving approximation migdal kadanov renormalisation as a result the author obtains a value smaller than the hausdorff dimension

flory calculation of the fractal dimensionality of the shortest path in a percolation cluster a flory type calculation of the fractal dimensionality of the shortest path in a percolation cluster near the threshold is introduced the result is roughly consistent with previous predictions or numerical simulations

on the accuracy of polynomial finite elements for crack problems a square tensile sheet with symmetric edge cracks is analysed by the finite element method to demonstrate the accuracy obtained with polynomial elements when the mesh patterns are chosen on the basis of the element s interpolation functions numerical results are provided for the accuracy of the total strain energy the stress intensity factor and the numerical condition of the global stiffness matrix the stress intensity factor is determined to within 1 0 per cent accuracy using quadratic linear strain triangular elements with only 138 degrees of freedom

growth model for fractal scale free networks generated by a random walk abstract the diversity of networked systems with fractal structures suggests that knowing the underlying mechanism that generates the fractality is necessary for building a model of the development of complex networks in the present paper we propose a growth model of a network generated by a random walk and show that the evolving graph forms a fractal structure with various properties including the scale free property if the graph which provides a space where a random walk occurs by itself is formed by the random walk the proposed model is regulated by two parameters p v and p e which define the probability of either a roundabout path via a new vertex or a shortcut being formed by the random walk respectively the power law exponent describing the vertex degree distribution is determined by the ratio p e p v and is related to an internal factor f i via the relation 1 f i 1 where f i is a parameter that describes the local structure generated by the random walk a sufficiently small p v provides the small world property to the model network the small world property is usually considered to be incompatible with the fractal scaling property open m c l c d c where open m c is the average number of vertices which can be reached from a randomly chosen vertex in at most l c steps however we demonstrate that fractality can be reconciled with the small world property by introducing a size dependent fractal cluster dimension d c

aggregation of particles which move on deterministic trajectories with fractal dimension two i a simple and new model for dla the author simulates a new type of aggregation process particles follow deterministic trajectories the trajectories have a fractal dimension dw of two dw is defined as s r approximately rdw where s r is the total number of sites a particle visits while travelling a distance r when the growth process is started with a single stationary particle on a square lattice a large aggregate looks like diffusion limited aggregation dla the fractal dimension df of the aggregate is nearly equal to 1 7 that of dla is 1 67 or 0 05 moreover the aggregate has axial anisotropy although it includes less than one thousand particles therefore the aggregation belongs to the same universality class of dla and is supposed to be the highly noise reduced model for dla because the laplace equation has no relationship to the aggregation process the laplace equation does not seem to be a necessary condition for producing aggregates like dla

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

scaling behaviour for excitation trapping on fractals the authors study the decay behaviour of a nearest neighbour random walker which gets trapped at the first encounter of a sink as underlying structures they consider fractals sierpinski gaskets and regular lattices and establish the scaling behaviour of the decay law with respect to x nd2 ln 1 p where n is the number of steps p the sink concentration and d the spectral dimension the decay scales well for x large for small x scaling is reasonable for the gaskets and the linear chain and rather poor for the square lattice

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

mean first passage time for diffusion on fractal lattices with imposed boundary conditions abstract relation between the mean first passage time and the chemical distance for diffusion on fractal lattices with imposed periodic boundary conditions is analysed heuristically it is argued that this relation is completely controlled by the chemical and the fractal dimensions and two parameters depending on the number of lattice sites according to the power law equations extensive computer simulations have been performed in order to confirm the analytical formulae

boundary element method for thermoelastic analysis of three dimensional transversely isotropic solids abstract thermal effects are well known to manifest themselves as additional volume integral terms in the direct formulation of the boundary integral equation bie for linear elastic solids when using the boundary element method bem this domain integral has been successfully transformed in an exact manner to surface ones only in isotropy and in 2d anisotropy thereby restoring the bem as a truly boundary solution technique the difficulties with extending it to 3d general anisotropic solids lie in the mathematical complexity of the green s function and its derivatives for such materials these quantities are required items in the bem formulation in this paper the exact analytical transformation of the volume integral associated with thermal effects to surface ones is achieved for a transversely isotropic material using a similar approach which the authors have previously employed for the same task in bem for 2d general anisotropy a numerical scheme however needs to be employed to evaluate some of the new terms introduced in the surface integrals that arise from this process here the mathematical soundness of the formulation is demonstrated by a few examples the numerical results obtained are checked by alternative means including those obtained from the commercial fem code ansys

random walks on ultrametric spaces low temperature patterns the authors present the numerically determined distribution of the number of sites visited by thermally activated walkers on ultrametric spaces which mimic the energetic disorder found in amorphous materials at high temperatures the results resemble those found for sierpinski type fractals at low temperatures the distributions are highly structured showing discontinuities which increase with the number of steps in contrast to an intuitive picture that would lead to smoothing they apply the results to the energy transfer in disordered systems and study the decay due to trapping

variance of random walks on cayley trees application to the trapping problem the authors investigate the trapping of particles on regularly multifurcating cayley trees by using random walk methods in order to find a good approximation for the survival probability of the walker they use a cumulant expansion and obtain an exact expression for the variance of the range of nearest neighbour random walks the expressions are corroborated to high accuracy by simulation calculations the method may turn out to be useful for more complex problems such as walks on ultrametric spaces

treatment of body force volume integrals in bem by exact transformation for 2 d anisotropic elasticity in the boundary element method bem based on the direct formulation body force effects manifest themselves as an additional volume integral term in the boundary integral equation bie the numerical solution of the integral equation with this term destroys the notion of the bem as atruly boundary solution method this paper discusses the treatment of this volume integral for two dimensional anisotropic elasticity with body forces present the analytical basis for transforming this integral exactly into boundary ones is presented for geometrically convex regions this restores the application of the bem to such problems as a truly boundary solution technique numerical examples are presented to demonstrate the veracity of the transformation and implementation 1997 by john wiley sons ltd

renormalisation on sierpinski type fractals presents a family of deterministic fractals which generalise the d dimensional sierpinski gaskets and establishes their order of ramification and their fractal and spectral dimensions random walks on these fractals are renormalisable and lead to rational not necessarily polynomial mapping

a recurrence formula for the direct solution of singular integral equations abstract an iterative method for the solution of singular integral equations is given in this paper by developing a recurrence formula discretizing the above formula by using appropriate quadrature rules the solution of the singular integral equation is given in an extremely simple form the number of numerical operations required for such a solution is considerably reduced when compared to the number of operations required for a classical type of solution illustrative examples are given indicating the efficiency of the method it is shown that the number of operations in this procedure is only half the number of the operations for a typical numerical method the convergence of the method is studied in the space of holder continuous functions in the particular case of plane elasticity more efficient bounds are given in the same case it is proved that the procedure is equivalent to the schwarz s alternating method and convergence is assured 18

random walks on ultrametric spaces mean and variance of the range the authors study thermally activated hopping processes on regularly multifurcating ultrametric spaces ums and investigate the first two moments of the range rn where n is the number of steps paralleling the recent findings for cayley trees the knowledge of exact generating functions for these moments permits to draw conclusions about their behaviour for large n for the first moment the mean number of distinct sites visited sn they derive analytically the cross over from a linear behaviour in the transient regime gamma 1 to a sublinear increase sn approximately ngamma in the recurrent regime gamma 1 here as usual for ums the parameter gamma depends linearly on the temperature the second moment of rn the variance vn displays different behaviour corresponding to strongly transient gamma 2 not strongly transient 1 gamma 2 and recurrent gamma 1 random walks all results are verified by comparison to computer data obtained through simulations and through numerical series expansions

scaling theory for the dynamic density profile at criticality a scaling description for the dynamic density profile of kinetic clusters formed by the sites visited during random walks executed on critical percolation clusters is proposed the analysis is done for both euclidean and fractal spaces various percolative diffusion laws are correctly recovered from this dynamic density profile

3d bem for general anisotropic elasticity abstract a new method for computing the system matrices g and h of the three dimensional boundary element method is presented for the elastostatic problems of general anisotropy green s functions expressed by the line integrals over a semi circle have a simple structure that allows analytical integration over the boundary elements the computation for the system matrices g and h is reduced to numerical integration of regular line integrals over the unit semi circle the proposed method has been implemented and tested with numerical examples

singular element construction using a mapping technique conformal mapping regularizes in the new plane the singular stress field around the crack tip so conventional finite elements can be used in the transformed plane the inverse mapping of the above elements gives another class of mapped elements the crack tip mapped element is a singular element thus using various elements the parent elements in the mapped plane we obtain an infinity of singular elements the product elements these singular elements are introduced naturally and have no defect as many other singular elements have

analysis of thermoelastic stress concentration around oblate cavities in three dimensional generally anisotropic bodies by the boundary element method abstract in this article stress concentrations around cavities in 3d generally anisotropic bodies are investigated using the boundary element method bem where the associated volume integral is analytically transformed to the boundary all derivations are based upon the fourier series representations of the fundamental solutions including green s function of displacements and its derivatives this approach of analytical transformation has fully restored the bem s distinctive notion that only the boundary needs to be discretized the goal of the present work is to investigate the thermoelastic stress concentration around cavities in 3d anisotropic bodies by use of the analytically transformed boundary integral equation bie the work has fully recovered the bem s nature of boundary discretization for treating 3d generally anisotropic thermoelasticity this is the first implemented work that successfully treats thermoelastic problems for 3d generally anisotropic solids by the analytically transformed bie in the paper interesting phenomena are observed from the analyses of stress concentrations around oblate cavities and some discussions are made

quantum field theory models on fractal spacetime continuing the analysis of a previous paper the present work applies rigorous renormalization group methods to the hierarchical models to establish the existence of field theories with non gaussian ultraviolet renormalization group fixed points in 4 e dimensions

spreading of percolation in three and four dimensions the authors present monte carlo simulations of the spreading of bond percolation on three and four dimensional simple hyper cubic lattices the algorithm is the same as that applied previously in two dimensions with the spreading proceeding from a hyperplane they find the spreading dimension to be d 1 82 or 0 02 three dimensional and d 1 88 or 0 03 four dimensional also they obtain values of the percolation probability and of the static exponents with errors at least comparable to the best values from the literature

numerical evidence of the critical percolation probability pc 1 for site problems on sierpinski gaskets the authors propose a new method the alternating translational dilation method atdm and use it in combination with the monte carlo technique to calculate the critical percolation probability for sierpinski gaskets sg they obtain numerical evidence of the critical percolation probability pc 1 for site problems on sg and an approximate relation pc 1 1 n in two dimensions where n is the number of stages of sg

a fast algorithm for the multiplication of generalized hilbert matrices with vectors on presente un algorithme rapide de complexite logarithmique pour la multiplication des matrices de hilbert generalisees par des vecteurs

random walk renormalisation and the spectral dimension of harmonic models on a class of hierarchical lattice random walk recursion relations are developed on a simple example of hierarchical lattice and are found to be the generating functions for first passage walks with exact renormalisation of a general harmonic model the spectral dimension f for the bond hierarchy class is shown to be f 2ln g ln g ln lambda r where g and lambda r are respectively the aggregation number and resistance eigenvalue the behaviour of f across families of hierarchies is discussed it is noted that the renormalisation group has limitations in its abilities to give results for the above models on hierarchies and that f so found may not maintain on hierarchies its usual significance for random walk statistics

nystro m s iterative variant methods for the solution of cauchy singular integral equations iterative methods for the solution of algebraic systems and integral equations are becoming increasingly popular because they can take advantage of the speed of vector and parallel computers in this paper nystrom s iterative variant methods developed for the solution of fredholm integral equations are extended to the solution of cauchy singular integral equations gauss chebyshev and lobatto chebyshev quadrature approximations are used to define two iterative algorithms giv and liv and their convergence for all continuously differentiable kernels is proven for arbitrary kernels the complexity is o m 2 operations per iteration step where m is the number of node points however for certain kernels the complexity can be reduced to o m log 2 m by using a fast summation algorithm an extrapolation technique based on the average of the gauss chebyshev and lobatto chebyshev methods is also introduced the average is usually as accurate as the lobatto chebyshev or gauss chebyshev method with twice

some general lagrange interpolations over simplex finite elements with reference to derivative singularities abstract for simplex finite elements the native lagrange family with arbitrarily placed nodes is presented in hierarchy ranking expressions it includes the well known complete lagrange family as well as the mid edge lagrange family to be proposed in this paper this new family enables us to utilize harmonious combinations of interpolations of different orders in finite element analysis as an application of developed simplex interpolations to fracture mechanics where some derivative singularities are needed we then describe the semi radial singularity mapping with examinations of peculiar trial function spaces

metric properties of fractal lattices the authors study the connectivity dimension d of fractal lattices viewed as networks graphs of sites and constant length bond two examples are investigated in detail the 2d sierpinski gasket and 2d infinite percolation clusters on square and triangular lattices in the first case d is shown to coincide with the fractal dimension whereas d 1 72 or 0 02 appears as a universal exponent for percolation clusters in two dimensions considered as an intrinsic parameter the connectivity dimension d is compared with other intrinsic and extrinsic characteristic parameters of fractal lattices in particular the authors argue that d or d or d holds on fractal lattices in general d fractal dimension d spectral dimension

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

the mapped elements for the solution of cracked bodies abstract using a conformal mapping the singular stress field around crack tips is transformed to a regular one with respect to the new variables thus the displacement field is approximated in the mapped plane the inverse mapping gives shape functions which are not polynomials the strain field obtained after a derivation of the displacement field has the appropriate square root singularity the elements constructed by using this procedure are called mapped elements this simple method was applied with excellent results to the solution of a series of problems

random walks on invasion percolation clusters results are presented of numerical simulations of random walks on invasion percolation clusters in two and three dimensions including simulations on continuum percolation clusters at threshold walks of up to 105 steps were taken the results indicate that the fracton dimension of invasion percolation clusters both on regular lattices and in the continuum is the same within numerical errors as the fracton dimension of ordinary percolation clusters at threshold

quadrilateral element with semi radial singularity mapping the singularity mapping elementology based on the constant and semi radial field conditions is presented in an originally quadrilateral finite element with the complete evaluations of singularities in the stress and strain as well as of the strain energy finiteness it is demonstrated that this singularity element family is highly versatile differences and some identities between the semi radial mapping and the so called quarter point mapping are also examined

transfer matrix calculation of the relative noise exponent in a two dimensional percolating network a new method is given for calculating the relative noise exponent describing the size dependence of macroscopic fluctuations due to microscopic resistance noise of a random resistor network at the percolation threshold pc the macroscopic fluctuation is calculated exactly using the transfer matrix idea in a given network the asymptotic exponent b is found by finite size scaling

subdiffusive continuous time random walks with stochastic resetting we analyze two models of subdiffusion with stochastic resetting each of them consists of two parts subdiffusion based on the continuous time random walk scheme and independent resetting events generated uniformly in time according to the poisson point process in the first model the whole process is reset to the initial state whereas in the second model only the position is subject to resets the distinction between these two models arises from the non markovian character of the subdiffusive process we derive exact expressions for the two lowest moments of the full propagator stationary distributions and first hitting time statistics we also show with an example of a constant drift how these models can be generalized to include external forces possible applications to data analysis and modeling of biological systems are also discussed

global first passage times of fractal lattices the global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time t we find simple expressions for the mean global first passage time for five fractals the d dimensional sierpinski gasket t fractal hierarchical percolation model mandelbrot given curve and a deterministic tree we also find an exact expression for the second moment and show that the variance of the first passage time var t scales with the number of nodes within the fractal n such that var t similar to n 4 d where d is the spectral dimension

dimension of discrete fractal spaces a new definition of fractal dimension in the case of discrete metric spaces is given it defines the dimension of an arbitrary unbounded subset x of the nu dimensional lattice zv

quantum field theory models on fractal spacetime i introduction and overview deux types de modeles de champs scalaires sont consideres l un a son propagateur donne par un temps propre ou une representation chaleur noyau l autre a un propagateur hierarchique la version du reseau fractal du propagateur a temps propre est une reflexion positive

diffusion on percolation clusters at criticality the concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters the authors find from monte carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three dimensional space is d 3 3 or 0 1 where the size of the cluster is restricted to be larger than the span of the walk and is d 3 9 or 0 1 for a walk on clusters not subject to this restriction for two dimensional space they find d approximately d approximately 2 7 or p0 1 the exponent d and d is related to the scaling of the average length r of n steps via rd varies as n the fracton dimensionality which is related to the density of states was found to be 1 26 or 0 1 these results are in good agreement with the predictions of alexander and orbach 1982

fractal chemical kinetics reacting random walkers computer simulations on binary reactions of random walkers a a a on fractal spaces bear out a recent conjecture p po i oct s where p is the instantaneous walker density and p0 the initial one andf d j2 where d s is the spectral dimension for the sierpinski gaskets d 2 2f 1 38 d 1 365 d 3 2f 1 56 d s 1 547 biased initial random distributions are compared to unbiased ones for site percolation d 2 p 0 60 2f 1 35 d 1 35 d 3 p 0 32 2f 1 37 d s 1 4 fractal to euclidean crossovers are also observed for energetically disordered lattices the effective 2f from reacting walkers and d s from single walkers are in good agreement in both two and three dimensions

single random walker on disordered lattices random walks on square lattice percolating clusters were followed for up to 2 10 steps the mean number of distinct sites visited sn gives a spectral dimension of d s 1 30 5 0 03 consistent with superuniversality d s 4 3 but closer to the alternative ds 182 139 based on the low dimensionality correction simulations are also given for walkers on an energetically disordered lattice with a jump probability that depends on the local energy mismatch and the temperature an apparent fractal behavior is observed for a low enough reduced temperature above this temperature the walker exhibits a crossover from fractal to euclidean behavior walks on two and three dimensional lattices are similar except that those in three dimensions are more efficient

diffusion and trapping of excitations in disordered systems we investigate diffusion and trapping of excitations in percolation systems by an effective medium approximation we calculate p0 t the probability of being at the origin at time t on the percolation cluster at the percolation threshold pc sn t the mean number of distinct sites visited after time t at and above pc and n t the survival probability after time t all of our findings are completely consistent with the previously proposed relations by rammal and toulouse we also consider a more general problem in which the waiting time distribution of an associated continuous time random walk is long tailed and propose generalisations of the rammal toulouse conjectures our results indicate that the scaling properties of these quantities are sensitive to the details of the system

some special interpolation formulae for triangular and quadrilateral elements presented herein are methods to generate special interpolation formulae of the following type i c0 continuous interpolation over triangular elements which induces a given type of singularity for the first derivatives at one of the vertices and yet preserves rigid and constant strain motions ii c1 continuous interpolation over triangular elements which induces a given type of singularity for second derivatives at one of the vertices

the tanh transformation for the solution of singular integral equations using a tanh transformation a quadrature formula for the evaluation of singular integrals is obtained the formula has the same step length h as the formula for regular integrals derived by f stenger these quadrature formulae are valid for end point singularities of any order and their error exhibits an exponential decay when the number of integrations tends to infinity using these formulae the solution of singular integral equations does not depend on the order of the end point singularities furthermore the collocation points are given by a very simple equation and in the case of constant coefficients by a closed form formula

surface and edge exponents for the spreading of 3d percolation monte carlo results are presented for the spreading of 3d percolation where the seed consists of an infinite straight line this line can either be in the bulk of the material into which spreading is possible on a planar surface of the material or on a rectangular edge in the last two cases spreading occurs only into angular regions with 180 degrees resp 90 degrees while the mean distance grows at p pc in all three cases with the same exponent the average number of growth sites grows resp vanishes in all three cases with different exponents in particular this implies that at p pc the sites on the surface of a big cube belonging to the maximal cluster within that cube have fractal dimension 1

shortest path fractal dimension for percolation in two and three dimensions we carry out a high precision monte carlo study of the shortest path fractal dimension d mathrm min for percolation in two and three dimensions using the leath alexandrowicz method which grows a cluster from an active seed site a variety of quantities are sampled as a function of the chemical distance including the number of activated sites a measure of the radius and the survival probability by finite size scaling we determine d mathrm min 1 130 phantom rule 0 16em 0ex 77 2 and 1 3756 6 in two and three dimensions respectively the result in two dimensions rules out the recently conjectured value d mathrm min 217 192 deng et al phys rev e 81 020102 r 2010

diffusion on random systems above below and at their percolation threshold in two and three dimensions a detailed monte carlo study is presented for classical diffusion random walks on random l l triangular and l l l simple cubic lattices with l up to 4096 and 256 respectively the speed of a cyber 205 vector computer is found to be about one order of magnitude larger than that of a usual cdc cyber 76 computer to reach the asymptotic scaling regime walks with up to 10 million steps were simulated with about 10 jl steps in total for l 256 at the percolation threshold we review and extend the dynamical scaling description for the distance traveled as function of time the diffusivity above the threshold and the cluster radius below earlier discrepancies between scaling theory and computer experiment are shown to be due to insufficient monte carlo data the conductivity exponent is found to be 2 0 0 2 in three and 1 28 0 02 in two dimensions our data in three dimensions follow well the finite size scaling theory below the threshold the approach of the distance traveled to its asymptotic value is consistent with theoretical speculations and an exponent 2 5 independent of dimensionality the correction to scaling exponent at pc seems to be larger in two than in three dimensions

on the spreading dimension of percolation and directed percolation clusters the authors study the number an of sites that are accessible after n steps at most on clusters at the percolation threshold on a cayley tree an is of order n2 if the origin belongs to a larger cluster whereas its average over all clusters is of order n this suggests that the intrinsic spreading dimension d defined by an approximately nd is equal to two for fractal percolation clusters in space dimensions d or 6 and depends on d for d 6 for directed percolation clusters they argue that d is related to usual critical exponents by d beta gamma nu sub monte carlo data that support this relation are presented in two dimensions analogous results are derived for lattice animals d 2 on the cayley tree and d 1 nu sub for directed animals in any dimensions

derivatives of the three dimensional green s functions for anisotropic materials abstract a concise formulation is presented for the derivatives of green s functions of three dimensional generally anisotropic elastic materials direct calculation for derivatives of the green s function on the cartesian coordinate system is a common practice which however usually leads to a complicated course in this paper the green s function derived by ting and lee ting t c t lee v g 1997 the three dimensional elastostatic green s function for general anisotropic linear elastic solids the quarterly journal of mechanics and applied mathematics 50 3 407 426 is extended to obtain the derivatives using a spherical coordinate system the green s function can be shown as the composition of two independent functions one depends only on the radial distance of the field point to the origin and the other is in spherical angles the method of derivation is based on the total differential scheme and then takes its partial differentiation accordingly with the application of cauchy residue theorem the contour integral can be evaluated in terms of the stroh eigenvalues of a sextic equation for the degenerate case evaluation of residues at multiple poles is also given applications of the present result are made to examine the green s functions and stress components for isotropic and transversely isotropic materials the results are in exact agreement with existing solutions

phase diagram for the spin 1 annni model using the method of ring recurrence the 2d spin 1 annni model is studied using an iterative method the method of ring recurrence the phase diagram is obtained numerically using matrices of order 9 the authors find the phase diagram to be generally similar to that of the 2d spin 1 2 annni model

analytical transformation of the volume integral in the boundary integral equation for 3d anisotropic elastostatics involving body force abstract in the boundary element method bem it is well known that the presence of body force shall give rise to an additional volume integral that conventionally requires domain discretization for numerical computations to restore the bem s distinctive notion of boundary discretization the present work analytically transforms the volume integral to surface ones for the body force effect in the 3d anisotropic elasticity on applying green s theorem new fundamental solutions with explicit forms of fourier series are introduced to facilitate the volume to surface transformation the coefficients of the fourier series representations are determined by solving a banded matrix formulated from integrations of the constrained equation of no doubt such an approach has fully restored the boundary element method as a truly boundary solution technique for analyzing 3d anisotropic elasticity involving body force at the end numerical verifications of the volume to surface integral transformation are presented also such an approach has been implemented in an existing bem code for demonstrating the implementation numerical examples are presented with comparisons with ansys analysis to the author s knowledge this is the first work in the open literature that reports the successful transformation for 3d anisotropic elasticity

random walk statistics on fractal structures we consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds range renewal theory mean first passage time etc asymptotic behaviors are shown to be controlled by the fractal d and spectral d dimensionalities of the considered structure a simple decimation procedure giving the value of d is outlined and illustrated in the case of the sierpinski gaskets recent results for the trapping problem the self avoiding walk and the true self avoiding walk are briefly reviewed new numerical results for diffusion on percolation clusters are also presented

the singular value decomposition of the gauss chebyshev and lobatto chebyshev methods for cauchy singular integral equations we obtain a closed form expression for the singular values and singular vectors of two matrices arising in the approximation of cauchy singular integral equations csie by the gauss chebyshev and lobatto chebyshev quadratures we also derive their singular value decomposition we use this decomposition to investigate the convergence of an iterative method for csie s proposed by ioakimidis an iterative algorithm for the numerical solution of singular integral equations j computational physics 43 164 176 1981 we show that this iterative method converges only under very restrictive conditions

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

true self avoiding walk on fractals the authors investigate the true self avoiding walk on fractal spaces the number sn of sites visited during an n step walk and the root mean square displacement rn are calculated via extensive monte carlo simulations on the 2d sierpinski gasket for all positive values of the repulsion parameter g they found the same exponents s 0 815 or 0 001 and nu 0 510 or 0 005 respectively both rn and sn are shown to exhibit a simple scaling behaviour in n and g the authors compare the results with a simple flory like prediction for the exponent nu

fundamental theory of the semi radial singularity mapping with applications to fracture mechanics abstract fundamental theory of the semi radial singularity mapping is presented which yields the necessary singularities in the stress and strain with bounded strain energy in particular the quadrilateral semi radial singularity mapping as well as the triangular one with collapse are described explicitly with some verification example in linear fracture mechanics the trial function spaces due to the conventional polynomial interpolations with these peculiar mappings are examined in detail