random walks on hierarchical lattices at the percolation threshold random walks on hierarchical lattices are considered at the percolation threshold real space renormalisation methods are employed to obtain the exact fractal dimension of the incipient infinite cluster iic df and the fractal dimension of the effective one dimensional resistance length of the iic dr for some hierarchical lattices the recent conjecture of aharony and stauffer does not hold for diamond and wheatstone bridge hierarchical lattices

exact analysis of the self avoiding random walks on two infinite families of fractals abstract we introduce two infinite families of fractals that we name the family and the koch family according to their first members which are the plane filling lattice and the koch fractal curve respectively the fractal dimension d f of family varies from 2 to 1 and from 1 465 to 1 in the case of the koch family when the fractal enumerator b an odd integer varies from 3 to we have calculated exactly the critical exponents of the self avoiding random walks saws on these fractals our results render it possible to perform a complete and exact study of the fractal to euclidean crossover which in this case occurs when b it turns out that all critical exponents when d f 1 b tend to the corresponding euclidean values with a unique correction term of the type constant ln b

critical exponents of self avoiding walks on a family of truncated n simplex lattices the authors have studied self avoiding walks saws on a family of truncated n simplex lattices which provide a family of fractals in which the fractal dimension d can be varied to a wide range while the spectral dimension d is held almost fixed by means of exact renormalisation group transformations they have calculated the critical exponents nu alpha and gamma and the connectivity constant mu of the saws for n 5 and 6 they propose an approximate theory for calculating the critical exponents of the saws on fractals which is expected to be accurate at large values of n they show that the theory gives results which are in good agreement with exact values even for small values of n and leads to simple closed relations for the exponents

persistence in nonautonomous predator prey systems with infinite delays this paper studies the general nonautonomous predator prey lotka volterra systems with infinite delays the sufficient and necessary conditions of integrable form on the permanence and persistence of species are established a very interesting and important property of two species predator prey systems is discovered that is the permanence of species and the existence of a persistent solution are each other equivalent particularly for the periodic system with delays applying these results the sufficient and necessary conditions on the permanence and the existence of positive periodic solutions are obtained some well known results on the nondelayed periodic predator prey lotka volterra systems are strongly improved and extended to the delayed case

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

speculations on self avoiding surfaces in fractals a mean field treatment the authors estimate the exponents characterising the self avoiding surfaces using an approximation in the framework of a flory type theory they find for planar self avoiding surfaces embedded randomly in a fractal of dimensionality d nu 3 4 d for random surfaces of fractal dimension d embedded in a euclidean space of dimensionality d nu 3 2d d 2 and for fractal surfaces embedded in a structure of fractal dimensionality d nu 3 2d d 2

geometrical arguments against the alexander orbach conjecture for lattice animals and diffusion limited aggregates geometrical arguments are given which suggest that the alexander orbach conjecture 1982 does not hold for lattice animals in d 2 and for the diffusion limited aggregates for large dimensions

stability conditions for systems of linear nonautonomous delay differential equations abstract systems of linear nonautonomous delay differential equations are considered which are of the form y i t k 1n 0t bik t s yk t s d ik s ci t yi t where i 1 n sufficient conditions are derived for both the asymptotic stability and the instability of the zero solution the main result is found by a monotone technique using elementary methods only moreover additional criteria are obtained by using the method of lyapunov functionals

self avoiding walks on random networks of resistors and diodes abstract we study the self avoiding walks saw on a square lattice whose various degrees of randomness encompasses many different random networks including the incipient clusters of the directed mixed and isotropic bond percolation we apply the position space renormalization group psrg method and demonstrate that within the framework of this method one is bound to find that the critical exponent v of the mean end to end distance of saw on various two dimensional random networks should be equal to the critical exponent of saw on the ordinary square lattice a detailed analysis of this finding and similar findings of other authors lead us to conclude that a debatable opposite finding which has been predicted on the basis of different approaches could be attained after a substantial refinement of the method applied

elastic properties of diffusion limited aggregates a transfer matrix method has been used to investigate the power law dependence of the effective elastic moduli of a two dimensional diffusion limited aggregate on the size l of the aggregate scaling exponents have been found to depend on the type of applied deformation for stretching along the direction of growth the elastic modulus scales as l 2 2 or 0 2 while for deformations in the direction perpendicular to the direction of growth it is proportional to l 2 8 or 0 2

renormalization and localization expansions ii expectation values of the fluctuation measures this paper is a continuation of 5 the k 1 st renormalization transformation applied to a density k is studied further the new contributions to the effective action obtained in 5 are renormalized and their localization expansions are constructed this completes proofs of theorems 1 and 4 in 5

the variational problems for classical n vector models we prove existence and regularity properties of solutions of the variational problems introduced in the previous paper 1 for classical lattice n vector models these results form a basis of our renormalization group approach to low temperature expansions for the considered models

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

on exact rates of decay of solutions of linear systems of volterra equations with delay this paper considers the resolvent of a finite dimensional linear convolution volterra integral equation the main results give conditions which ensure that the exact rate of decay of the resolvent can be determined using a positive weight function related to the kernel the decay rates can be exponential or subexponential many other related results on exact rates of exponential and subexponential decay of solutions of volterra integro differential equations are given we also present an application to a linear compartmental system with discrete and continuous lags

self avoiding walks on fractals studied by the monte carlo renormalization group the authors have applied the monte carlo renormalization group mcrg method to study the problem of self avoiding walks on the sierpinski gasket family of fractals each member of the family is labelled by an integer b 2 or b or infinity and when b to infinity both the fractal df and spectral ds dimension approach their euclidean value 2 they have calculated the critical exponent nu associated with the mean square end to end distance up to b 80 their mcrg results deviate at most 0 03 from the available exact results for 2 or b or 9 the obtained data show clearly that nu monotonically decreases with b and crosses the euclidean value nu 3 4 at b approximately 27 that is before entering the fractal to euclidean crossover region that occurs in the limit b to infinity

self avoiding random walks on multifractal lattices a renormalisation theory is developed to study the critical behaviour of self avoiding random walks on multifractals critical exponents and connectivity constants are calculated for walks on a class of square multifractal lattices using a finite lattice renormali sation the effect of the multifractal disorder is considered for both annealed and quenched disorder

superuniversal spectral dimension for dilute branched polymers the authors introduce the problem of random walks on lattice animals dilute branched polymers and calculate the spectral dimension ds of animals in two and three dimensions by using a two parameter position space renormalisation group method this problem which is a generalisation of the problem of the ant in the labyrinth has been independently proposed and investigated by wilke et al 1984 the spectral dimension is given by ds 2da dw where da is the fractal dimension of the lattice animal and dw the fractal dimension of the random walk on the lattice animal the results indicate that ds may be a superuniversal quantity i e its value is independent of dimension moreover the authors find ds to be close to 4 3 which is the spectral dimension of the largest percolation cluster at the percolation threshold pc as conjectured by alexander and orbach 1982 since both lattice animals and the largest percolation cluster at pc have a homogeneous interior structure the results suggest that the spectral dimension of all such fractals may equal 4 3

on the position space renormalisation group approach to diffusion limited cluster growth problems the authors investigate the application of position space renormalisation group methods to the witten sander 1981 model of diffusion limited cluster growth they show that this method cannot be used to investigate the asymptotic behaviour of the witten sander model in three and higher dimensions even though it yields satisfactory results in two dimensions with small cells

critical exponents of the self avoiding walks on a family of finitely ramified fractals the authors have studied the self avoiding walks saw on a family of finitely ramified fractals the first member b 2 of the family is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the homogeneous triangular lattice by means of the exact renormalisation group transformations they have calculated the critical exponents alpha nu and gamma and the connectivity constant mu of saw on each member of a sequence 2 or b or 8 of the studied fractal family the obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two dimensional lattice

potts model formulation of branched polymers in a solvent a potts model formulation of the statistics of branched polymers or lattice animals in a solvent is given the migdal kadanoff renormalisation group is employed to study the critical behaviour or fractal dimension of the branched polymer four different critical behaviours are found corresponding to random animal collapse or theta point percolation and compact cluster the theta point behaviour is described by a tricritical point while percolation corresponds to a higher order critical point where the effect of the solvent on the branched polymer is the same as the screening effect of the other clusters in percolation

a variation of constants formula for abstract functional differential equations in the phase space abstract for linear functional differential equations with infinite delay in a banach space a variation of constants formula is established in the phase space as an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line

global stability for infinite delay lotka volterra type systems abstract in this paper sufficient conditions are established for the global stability of the saturated equilibrium of an infinite delay nonautonomous lotka volterra type system the present work is distinguished from previous work principally be allowing the system to be nonautonomous and relaxing the traditional requirement that the undelayed intraspecific competition dominates both the delayed intraspecific competition as well as the interspecific interactions we require the undelayed intraspecific competition to dominate the latter but not the former

a low temperature expansion for classical n vector models iii a complete inductive description fluctuation integrals we give a complete inductive description of the effective densities generated by applications of the renormalization transformations to the densities of the classical n vector models described in the paper 1 and we formulate the main theorems of this sequence of papers this description includes now all the large field domains which we ve excluded from the discussions of 1 4 we apply the renormalization transformation to a density satisfying the inductive hypotheses we introduce large field small field partitions and we determine and analyze the fluctuation integrals in the general setting

anomalous diffusion on regular and random models for diffusion limited aggregation families of regular fractal models for witten sander dla clusters in d dimensions are proposed resembling the computer generated aggregates the authors evaluate random walk dimensionalities exactly using resistance scaling ideas the applicability of the conflicting conjectures of alexander orbach 1982 ao dw 3df 2 and aharony stauffer 1984 as dw df 1 to dla is then examined one finds that the dendritic nature of the fractals presented ensures that as is satisfied exactly in all dimensions an rsrg fugacity transformation method is applied to the simplest of the regular fractals in d 2 the result for dw is compared with the exact resistance scaling result to obtain an estimate of the error introduced by the rsrg the rg method is then generalised to treat the random dla problem the resulting estimate for dw is in fortuitously good agreement with as the corresponding monte carlo data and the results above for the regular fractal models

boundedness and asymptotic stability for delayed equations of logistic type for a scalar lotka volterra type delay equation x t b t x t 1 l x t where l c r 0 r r is a bounded linear operator and b a positive continuous function sufficient conditions are established for the boundedness of positive solutions and for the global stability of the positive equilibrium when it exists special attention is given to the global behaviour of solutions for the case of l a positive linear operator the approach used for this situation is applied to address the global asymptotic stability of delayed logistic models in the more general form x t b t x t a t l t x t with l t being linear and positive

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

exact series studies of self avoiding walks on two dimensional critical percolation clusters the author presents results of exact series studies of self avoiding walks on percolation clusters on the square lattice performed very close to the percolation threshold and estimates the critical exponents nu and gamma defined by the disorder averages of the mean square radius of gyration or the end to end distance and the number of self avoiding walks respectively the self avoiding walk configurations are enumerated exactly up to 35 steps the result for nu indicates a large increase compared with the self avoiding walks on fully occupied lattice the result for gamma however indicates a behaviour similar to that on a fully occupied lattice the result for nu is completely different from that of lee nakanishi and kim 1989

two dimensional diffusion limited aggregation a finite size scaling approach finite size scaling in the diffusion limited aggregation model of witten and sander 1981 1983 is studied in two dimensions via monte carlo simulations on semi infinite strips of width n 2 20 due to a crossover between self avoiding walk like growth for small n and dendritic growth for large n wide strips n 8 20 are needed to deduce the witten sander fractal dimension d approximately 5 3 from the finite size scaling analysis

universality in branched polymers on d dimensional hypercubic lattices the authors have derived series for weakly and strongly embeddable trees in d dimensional simple hypercubic lattices for arbitrary integral d for d 2 3 9 they present series evidence that such trees are in the same universality class as lattice animals in addition they have derived expansions in inverse powers of sigma 2d 1 for the growth parameters for bond and site trees and compare these with the corresponding results for animals

regularity and decay of lattice green s functions in the paper we study a class of lattice covariant laplace operators with external gauge fields we prove that these operators are positive and that their green s functions decay exponentially they also have regularity properties similar to continuous space green s functions all the bounds are uniform in the lattice spacing

explicit estimates and limit formulae for the solutions of linear delay functional differential systems with nonnegative volterra type operators abstract in this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative volterra type operators in a recent paper of us a variation of constants formula has been given for such systems and this is particularly suited to handling the studied problems first we investigate the boundedness of the solutions then some estimates are given for the upper and lower limits of the solutions by using this we study the existence of the limit of the solutions and obtain a formula for the limit the applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities synchronisation and lotka volterra type delay functional differential systems

cluster renormalisation study of site lattice animals in two and three dimensions the author studies site lattice animals using the cluster renormalisation cr approach previously applied to bond animals the exponent v characterising the asymptotic behaviour of the mean square radius of gyration of site animals and the fractal dimension df 1 v are determined in two and three dimensions he finds v 0 649 or 0 009 and v approximately 0 51 in two and three dimensions respectively in addition he studies restricted valence site animals on a triangular lattice using a single parameter cr approach

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

diffusion in concentrated lattice gases the diffusion of many particles on a lattice is an example of a correlated random walk process recently the waiting time distributions for two consecutive jumps of a tagged particle have been determined numerically and the time dependent correlations analyzed in detail the information over these two consecutive jumps is used to determine the position and velocity autocorrelation functions and very satisfactory results are obtained for three dimensional lattices however the information of the two consecutive jumps is insufficient when the jump rate of the tagged particle is large compared to that of the other particles and this approximation fails completely in one dimension for the linear chain another approximation which accounts for correlations over all jumps is compared with the numerical simulations

on the relationship between the critical exponents of percolation conductivity and static exponents of percolation the author argues that the critical exponent t of random conductance networks near the percolation threshold is given by t d 1 nu for low dimensionalities and t 1 beta for high dimensionalities where nu is the correlation length exponent beta the backbone exponent and d is dimensionality the author argues that what separates the two regimes is a critical fractal dimensionality dl which equals 2 the author also argues that dl is also a critical fractal dimensionality for fractals such as lattice animals and diffusion limited aggregates the result for low dimensionalities has been also obtained by aharony and stauffer by a different argument

constructive field theory and applications perspectives and open problems in this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory they could give work for future mathematical physicists trained with constructive methods well into the 21st century

the large field renormalization operation for classical n vector models a renormalization operation of the large field densities for the classical n vector models is constructed in this paper it allows to improve bounds of these densities and thus to prove all inductive hypotheses this completes the construction and analysis of the full renormalization group flow for these models the results will be used in the next paper to analyze the correlation functions

a note on permanence of nonautonomous cooperative scalar population models with delays for a large family of nonautonomous scalar delayed differential equations used in population dynamics some criteria for permanence are given as well as explicit upper and lower bounds for the asymptotic behavior of solutions the method described here is based on comparative results with auxiliary monotone systems in particular it applies to a nonautonomous scalar model proposed as an alternative to the usual delayed logistic equation

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

possible relations for topological and transport properties of the lattice animal model of branched polymers the author proposes that the following relations hold exactly for the lattice animal model of branched polymers the resistivity exponent zeta a is exactly equal to da the fractal dimensionality of the backbone of the anomals the random walk fractal dimensionality dw is given by dw da da where da is the fractal dimensionality of the animals the spectral dimension ds of the backbone of the animals is given by ds 1 at all dimensions i e the backbone is a highly decorated but essentially chain like object these are in contrast with earlier suggestions that one needs two three topological properties to obtain zeta a dw the author proposes that these relations may hold for all clusters for which loops are irrelevant and have a finite upper critical dimensionality the implications of these results for diffusion limited aggregates and percolation clusters are also discussed

first exit time of termites and random super normal conductor networks the resistance of a random super normal conductor network is calculated through the diffusion of a termite which performs a random walk on the normal bonds and has the same probability to exist from any point on a cluster of superconducting bonds the first time t that the termite exits at a distance r from the origin averaged over many configurations and runs is found to behave as t approximately r2k for r xi pc p s describes the divergence of the conductivity as p to pc p being the concentration of the superconductor scaling arguments are used to show that k 1 s 2 theta where 2 theta is the fractal dimension of random walks on single clusters at pc numerical simulations at two dimensions d 2 yields s 1 34 or 0 10 and k 1 34 or 0 03 in agreement with scaling the authors also show that the probability of return to the origin at time t behaves as t dk2 preliminary results at d 1 and other calculations methods are also discussed

the renormalization group according to balaban ii large fields this is an expository account of balaban s approach to the renormalization group the method is illustrated with a treatment of the ultraviolet problem for the scalar 4 model on a toroidal lattice in dimension d 3 in this paper we control the large field contribution to the partition function

where two fractals meet the scaling of a self avoiding walk on a percolation cluster the scaling properties of self avoiding walks on a d dimensional diluted lattice at the percolation threshold are analyzed by a field theoretical renormalization group approach to this end we reconsider the model of y meir and a b harris phys rev lett 63 2819 1989 and argue that via renormalization its multifractal properties are directly accessible while the former first order perturbation did not agree with the results of other methods our analytic result gives an accurate description of the available mc and exact enumeration data in a wide range of dimensions 2 ensuremath leqslant d ensuremath leqslant 6

exact fractals with adjustable fractal and fracton dimensionalities a family of exact fractals is presented by adjusting two external parameters a wide range of fractal and fracton dimensionalities can be achieved this includes the case of fracton dimensionality of 2 which is critical for diffusion

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

localization expansions i functions of the background configurations expansions of the type described in the inductive hypothesis h 5 in the paper 1 are constructed for local functions of the background configurations i e solutions of the variational problems studied in the previous paper 3 a main part of this construction is a further analysis of a local structure of the solutions

on a delay population model with a quadratic nonlinearity without positive steady state a population model described by a nonlinear delay differential equation with a quadratic nonlinearity x t k 1 m k t x h k t t x 2 t t 0 is considered where m 1 is an integer functions k 0 0 are continuous functions h k 0 r are continuous such that t h k t t const 0 and for any t 0 the inequality h j t t holds for at least one index j 1 m although this equation does not have a positive steady state a new method not based on the existence of a positive steady state is developed and used to investigate the permanence global attractivity conditions and nonoscillation properties

diffusion and fracton dimensionality on fractals and on percolation clusters the authors present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions the results for the fractal support the rammal and toulouse proposition 1983 that ds n dn varies as b n s n the results for percolation are in excellent agreement with the alexander and orbach conjecture 1982 that the fracton dimensionality d 4 3

self avoiding walks on percolation clusters the author studies self avoiding walks saws on percolation clusters a scaling function representation for r the mean end to end distance is proposed which describes a crossover from ordinary saws to saws on fractals he distinguishes between saws on a single cluster for which r approximately nx where x nu 1 and saws on all clusters for which r approximately nz where z nu 2 and n is the number of monomers in the walk he estimates nu 1 d 2 approximately 1 285 and the correction to scaling exponent omega d 2 approximately 1 3 and nu 1 d 3 approximately 1 38 two plausible generalisations of the flory 1953 approximation for nu 1 are investigated and it is argued that none of them provides a satisfactory approximation for nu 1 at all dimensions

a low temperature expansion for classical n vector models ii renormalization group equations this paper continues the analysis of the low temperature expansions for classicaln vector models started in 1 a main part of it is a derivation of renormalization group equations and a construction of their solutions to do this we have to introduce a fluctuation integral connected with a next renormalization transformation and to make its preliminary analysis the results of the paper are summarized in theorems stating that the renormalization transformation preserves the space of densitites or actions described inductively in 1

diffusion on random clusters and the parasite problem in the parasite problem a particle ant diffuses randomly on a random percolation cluster in the limit of concentration to 0 lattice animal monte carlo simulations and scaling arguments show that for large animals the distance r travelled by this parasite increases as t 1 za with time t the authors find za approximately 3 4 on the simple cubic lattice and za approximately 2 6 on the square lattice this anomalous diffusion is in rough agreement with a generalisation of a suggestion by alexander and orbach 1982 heuristic arguments in favour of this suggestion are given also they look at corrections to scaling for concentrations equal to the percolation threshold

the backbone and conductivity of random clusters the author studies the properties of the backbone of and conduction in random clusters which have homogeneous interior structure these include the largest percolation cluster at pc lattice animals and the witten sander ws aggregates for lattice animals the fractal dimension da of the backbone is estimated the for first time and is found to be about 1 14 at d 2 and about 1 39 at d 3 these values are much lower than the corresponding values for the animals themselves the fractal and spectral dimensions of the backbone of the ws aggregates are estimated to be about 1 25 and 1 06 at d 2 respectively a recent hypothesis which relates the fractal dimension of random walks on these clusters to those of the clusters and their backbones is also discussed it is shown that for the ws aggregates ds ds 2 on a bethe lattice i e at d infinity in contrast with ds 4 3 and ds 1 for percolation clusters and lattice animals the conductivity exponent ta of lattice animals is found to be ta d 2 approximately 0 73 and ta d 3 approximately 1 19 and ta d or 8 2 whereas for the ws aggregates the author finds tws d 2 approximately 0 67 and tws d 3 approximately 0 94 and tws d to infinity 1

scaling theory for the statistics of self avoiding walks on random lattices the authors study self avoiding walks saw on randomly diluted quenched lattices with direct configurational averaging over the moments of the saw distribution function a scaling function representation of rn the average end to end distance of n step walks is studied here both for saw on a the infinite percolation cluster and b any cluster they have shown that at the percolation threshold nu p nu p 1 beta p 2 nu p where beta p and nu p are the percolation order parameter and correlation length exponents respectively the authors also propose a scaling function representation for the total number of n step saw configuration gn approximately mu nngamma 1 for infinite cluster averaging which gives gamma p gamma d nu p nu for all cluster averaging gamma will remain unchanged

periodic solutions of periodic delay lotka volterra equations and systems abstract by using the continuation theorem of coincidence degree theory sufficient and realistic conditions are obtained for the existence of positive periodic solutions for both periodic lotka volterra equations and systems with distributed or state dependent delays our results substantially extend and improve existing results