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

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

global attractor of a coupled finite difference reaction diffusion system with delays abstract in the study of asymptotic behavior of solutions for reaction diffusion systems an important concern is to determine whether and when the system has a global attractor which attracts all positive time dependent solutions the aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady state system applications are given to three types of lotka volterra reaction diffusion models where time delays may appear in the opposing species

random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

dimer monomer model on the sierpinski gasket abstract we present the numbers of dimer monomers m d n on the sierpinski gasket s g d n at stage n with dimension d equal to two three and four the upper and lower bounds for the asymptotic growth constant defined as z s g d lim v ln m d n v where v is the number of vertices on s g d n are derived in terms of the results at a certain stage as the difference between these bounds converges quickly to zero as the calculated stage increases the numerical value of z s g d can be evaluated with more than a hundred significant figures accurate from the results for d 2 3 4 we conjecture the upper and lower bounds of z s g d for general dimension the corresponding results on the generalized sierpinski gasket s g d b n with d 2 and b 3 4 are also obtained

existence theorems for certain elliptic and parabolic semilinear equations existence theorems for the nonlinear parabolic differential equation u t u u p f x t 0 in rn 0 with zero initial value are established given explicit conditions on the nonhomogeneous termf x t an existence theorem is also demonstrated for the corresponding elliptic equation

a higher order compact adi method with monotone iterative procedure for systems of reaction diffusion equations this paper is concerned with an existing compact finite difference adi method published in the paper by liao et al 2002 3 for solving systems of two dimensional reaction diffusion equations with nonlinear reaction terms this method has an accuracy of fourth order in space and second order in time the existence and uniqueness of its solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear reaction terms the convergence of the finite difference solution to the continuous solution is proved an efficient monotone iterative algorithm is presented for solving the resulting discrete system and some techniques for the construction of upper and lower solutions are discussed an application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method

free energy of magnetic model systems at the fractal to euclidian crossover abstract we prove that the free energy per spin of the nearest neighbour classical n vector spin model situated on the sierpinski gasket sg family of fractals tends to the free energy per spin of the same model situated on the standard triangular lattice when the linear size b of the sg fractal generator tends to infinity this finding can be regarded as the first step in the analytical investigation of the spin system thermodynamics at the fractal to euclidean crossover

existence and asymptotic behavior for elliptic equations with singular anisotropic potentials abstract we study the equation u u u p 1 v x u f x 0 in r n where n 3 and p n n 2 the forcing term f and the potential v can be singular at zero change sign and decay polynomially at infinity we can consider anisotropic potentials of form h x x 2 where h is not purely angular we obtain solutions u which blow up at the origin and do not belong to any lebesgue space l r also u is positive and radial in case f and v are asymptotic stability properties of solutions their behavior near the singularity and decay are addressed

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

qualitative analysis of a coupled reaction diffusion model in biology with time delays abstract certain biochemical reaction can be modeled by a coupled system of time delayed ordinary differential equations and linear parabolic partial differential equations in a three compartment model these equations are coupled through the boundary conditions the aim of this paper is to give a qualitive analysis of this unusual coupled system the analysis includes the existence and uniqueness of a global solution explicit upper and lower bounds of the solution and global stability of a steady state solution the global stability result is with respect to any nonnegative initial perturbation and is independent of the time delays in the process of reaction special attention is given to the goodwin model for biochemical control of genes by a negative feedback mechanism with time delay and diffusion

surface adsorption of a self avoiding polymer chain on a family of finitely ramified fractals we study the surface adsorption of a flexible self avoiding polymer chain on a family of sierpinski gasket type fractals using the real space renormalization group technique the members of this family are characterized by an integer scale factor b which runs from 2 to infinity the hausdorff dimension of these fractals tends to 2 from below as b to infinity we calculate the crossover exponent phi for the desorption transition exactly for b 2 to 6 for b to infinity we use finite size scaling arguments to show that phi to 1 4

monotone iterative methods for finite difference system of reaction diffusion equations this paper presents an existence comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary value problems the basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system using upper and lower solutions as two distinct initial iterations two monotone sequences from a suitable linear system are constructed it is shown that these two sequences converge monotonically from above and below respectively to a unique solution of the nonlinear discrete equations this formulation leads to a well posed problem for the nonlinear discrete system applications are given to several models arising from physical chemical and biological systems numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences

combined iterative methods for numerical solutions of parabolic problems with time delays we study monotone iterative schemes for finite difference solutions of reaction diffusion systems with time delays in this paper we derive two modified iterative schemes by combing the method of upper lower solutions and the jacobi method or the gauss seidel method the convergence and stability of the monotone iterative schemes are also discussed

bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity we consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter as the parameter crosses some critical values there appear certain resonances in the equation producing solutions that bifurcate from infinity we study the bifurcation branches characterize when they are sub or supercritical and analyse the stability type of the solutions furthermore we apply these results and techniques to obtain landesman lazer type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti maximum principle

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

residual entropies of the ising antiferromagnets on fractals ii d dimensional checkerboard type of fractals abstract we study the residual entropies of the antiferromagnetic ising systems in the maximum critical field situated on checkerboard cb type of fractals embedded in d dimensional euclidean space for a given d the cb fractals constitute a family whose each member is labelled by b where b is an odd integer 3 b

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

harmless delays for permanence in a class of population models with diffusion effects abstract this paper studies a class of time delay reaction diffusion systems modeling the dynamics of single or interacting populations in the logistic equation we prove that when the magnitude of the instantaneous term is larger than that of the delay terms the population growthuhas the same asymptotic limit as in the case of no delay for the predator prey model a condition on the interaction rates is given to ensure the permanence effect in the ecosystem regardless of the length of delay intervals a permanence condition is also obtained in then species competition system with time delays it is shown that when the natural growth rate a1 a2 an is in an unbounded parameter set the reaction diffusion system has a positive global attractor finally long term behavior of the solutions for those time delay systems is numerically demonstrated through finite difference approximations and compared with the corresponding systems without delays

higher order compact finite difference method for systems of reaction diffusion equations this paper is concerned with a compact finite difference method for solving systems of two dimensional reaction diffusion equations this method has the accuracy of fourth order in both space and time the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term three monotone iterative algorithms are provided for solving the resulting discrete system efficiently and the sequences of iterations converge monotonically to a unique solution of the system a theoretical comparison result for the various monotone sequences is given the convergence of the finite difference solution to the continuous solution is proved and richardson extrapolation is used to achieve fourth order accuracy in time an application is given to an enzyme substrate reaction diffusion problem and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach

an inhomogeneous semilinear equation in entire space abstract necessary conditions and a priori estimates are obtained for the inhomogeneous semilinear equation u up f x 0 in r n also several existence theorems are proved given explicit conditions on the inhomogeneous termf

multiple existence of solutions for a nonhomogeneous elliptic problem with critical exponent on rn abstract let n 3 2 2 n n 2 our purpose in this paper is to consider the existence and multiplicity of solutions of the problem u u 2 2 u f on r n u h 1 r n where f l 2 2 1 r n l r n f 0 and f 0

traveling wavefronts for a time delayed non newtonian filtration equation abstract this paper is concerned with traveling wavefronts for the evolutionary p laplacian with time delay we first establish some necessary and sufficient conditions for the existence of monotone nondecreasing and nonincreasing traveling wavefronts which correspond to definite wave speeds c 1 and c 2 respectively moreover special attention is also paid to the asymptotic behavior of traveling fronts connecting two uniform steady states

self avoiding walks on sierpinski carpets the author calculates the fractal dimensions of self avoiding walks saw on sierpinski carpets with the bond moving type renormalisation the results suggest that carpets with the same fractal dimension of the random walk have almost the same fractal dimension of saw within the approximation

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

tutte polynomial of pseudofractal scale free web the tutte polynomial of a graph is a 2 variable polynomial which is quite important in both combinatorics and statistical physics it contains various numerical invariants and polynomial invariants such as the number of spanning trees the number of spanning forests the number of acyclic orientations the reliability polynomial chromatic polynomial and flow polynomial in this paper we study and obtain a recursive formula for the tutte polynomial of pseudofractal scale free web psfw and thus logarithmic complexity algorithm to calculate the tutte polynomial of the psfw is obtained although it is np hard for general graph by solving the recurrence relations derived from the tutte polynomial the rigorous solution for the number of spanning trees of the psfw is obtained therefore an alternative approach to determine explicitly the number of spanning trees of the psfw is given furthermore we analyze the all terminal reliability of the psfw and compare the results with those of the sierpinski gasket which has the same number of nodes and edges as the psfw in contrast with the well known conclusion that inhomogeneous networks e g scale free networks are more robust than homogeneous networks i e networks in which each node has approximately the same number of links with respect to random deletion of nodes the sierpinski gasket which is a homogeneous network as our results show is more robust than the psfw which is an inhomogeneous network with respect to random edge failures

asymptotic form of the spectral dimension of the sierpinski gasket type of fractals the authors have studied the spectral dimension d48t of an infinite class of fractals the first member b 2 of the class is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the ordinary triangular lattice by studying the electric resistance of the fractals they have been able to calculate exact values of d for the first 200 members of the class an analysis of the obtained data reveals that for large b the spectral dimension should approach the upper limit of 2 according to the formula d approximately 2 constant ln b beta where beta is not larger than one this result implies among other things that the scaling exponents of the resistivity and diffusion constant should logarithmically vanish at the fractal lattice crossover

residual entropies of the ising antiferromagnets on fractals i d dimensional sierpinski gasket type of fractals we apply a new method to study the problem of the residual entropies of the antiferromagnetic ising systems in the maximum critical field situated on sierpinski gasket sg type of fractal lattices the sg fractal lattices comprise a family so that each member is labelled by an integer b 2 b and is embedded in d dimensional euclidean space in the two dimensional 2d and three dimensional 3d case we find expressions for the residual entropy b for arbitrary b and show that in the limit b the calculated values should converge to the values pertinent to the corresponding euclidean lattices in particular these results confirm recent findings concerning the residual entropy of the 2d sierpinski gasket type of fractals obtained by numerical fitting we also study the d dimensional case and find a general expression for b for arbitrary b 3

on nonlinear coupled reaction diffusion equations this paper is concerned with the existence and uniqueness of solutions of reaction diffusion equations arising in many applications and often coupled our approach to the problem is based on converting the coupled equations into a system of volterra fredholm type integral equations by means of the green s function representation and using the banach fixed point theorem an application to nonlinear second order evolution equations is also indicated

bounded solutions of delay nonlinear evolutionary equations we consider the initial value problem d u d t a u t f u t u t w t 0 u t t w t 0 in a banach space e with the positive operator a theorem on the existence and uniqueness of a bounded solution of this problem is established for a nonlinear evolutionary equation with time delay the application of the main theorem for four different nonlinear partial differential equations with time delay is shown the first and second order of accuracy difference schemes for the solution of one dimensional nonlinear parabolic equation with time delay are presented numerical results are provided

the ice model and the eight vertex model on the two dimensional sierpinski gasket abstract we present the numbers of ice model configurations with boltzmann factors equal to one i n on the two dimensional sierpinski gasket s g n at stage n the upper and lower bounds for the entropy per site defined as lim v ln i n v where v is the number of vertices on s g n are derived in terms of the results at a certain stage as the difference between these bounds converges quickly to zero as the calculated stage increases the numerical value of the entropy can be evaluated with more than a hundred significant figures accuracy the corresponding result of the ice model on the generalized two dimensional sierpinski gasket s g b n with b 3 is also obtained and the general upper and lower bounds for the entropy per site for arbitrary b are conjectured we also consider the number of eight vertex model configurations on s g n and the number of generalized vertex models e b n on s g b n and obtain exactly e b n 2 2 b 1 b b 1 2 n b 4 b 2 it follows that the entropy per site is lim v ln e b n v 2 b 1 b 4 ln 2

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

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

hopping on hierarchical structures and random walking on deterministic fractals the authors demonstrate that there exists a definite mapping between hopping on hierarchical structures with an arbitrary number of sons and random walking on one new infinitely large family of deterministic loopless fractals in addition by explicit calculation of the corresponding spectral dimensions of the introduced fractals they corroborate the validity of the einstein diffusion relation for inhomogeneous media

spanning trees on the sierpinski gasket we present the numbers of spanning trees on the sierpinski gasket sg d n at stage n with dimension d equal to two three and four the general expression for the number of spanning trees on sg d n with arbitrary d is conjectured the numbers of spanning trees on the generalized sierpinski gasket sg d b n with d 2 and b 3 4 are also obtained

fractal dimensionality of polymer chains the authors suggest that the concept of fractal dimensionality provides a useful characterisation of the configurational properties of a single polymer from numerical studies of long polymers traced out by self avoiding walks on a planar lattice they find that the fractal dimensionality is well defined and has a constant value for most scales of length of the chain renormalisation group theory provides a theoretical basis for the concept of fractal dimensionality in polymers

mandelbrot sets for pairs of affine transformations in the plane for a contractive matrix transformation a r2 to r2 a aij aij in r the set of points a or 1 or a or a2 or a3 or e for all sequences of and where e 1 0 t is a unique attractor of generally fractional hausdorff dimension the mandelbrot set associated with this system is defined as d aij in r4 a is contractive a is disconnected the structure of d and its boundary is investigated computer approximations of various sections of d are presented with a discussion of the algorithm and principles involved

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

comparison between some explicit and implicit difference schemes for hamilton jacobi functional differential equations abstract initial boundary value problems of dirichlet type for first order partial functional differential equations are considered explicit difference schemes and implicit difference methods are investigated sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented it is proved that assumptions on the regularity of given functions are the same for both the methods it is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods there are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent error estimates for both the methods are constructed numerical examples are presented

renormalisation on symmetric fractals the authors introduce and investigate new classes of sierpinski type fractals they determine their fractal and spectral dimensions using renormalisation procedures and for particular classes they give these dimensions in closed form the spectral dimensions densely fill the interval allowing them to flexibly choose models for applications

semigroups and nonlinear evolution equations abstract in the main part of this paper we survey some recent results on a semigroup approach to parabolic equations under nonlinear boundary conditions in particular we give a detailed motivation and a heuristic derivation of a generalized variation of constant formula which is the basis for the whole method in the appendix we prove some new results about semigroups in interpolation and extrapolation spaces

thermodynamical formalism for an infinite hierarchy of fractal resistor networks where n is the number of resistors we find a generic phase transition at 3 3c 1 also we develop a transfer matrix technique which determines all even positive moments the thermodynamical formalism is applied to the hilfer blumen hierarchy of generalized sierpinski gasket fractal networks and the crossover from fractal to lattice behavior is studied at this crossover we find a sharp phase transition in the second moment 3 2

topological frustration and quasicompact phase in a model of interacting polymers the authors study the geometric and thermal properties of a model of self interacting linear polymers self avoiding walks on a fractal lattice the three dimensional modified sierpinski gasket of base b 3 as a consequence of the topological structure of this lattice the polymer is frustrated and cannot fill the available space with a finite density when strong attractive interactions between monomers are present the polymer cannot reach the compact globule state usually observed rather it shrinks into a novel phase the semicompact state below a finite critical temperature in this phase the monomer density per lattice site vanishes asymptotically for large polymers the authors give the exact values of the critical exponents at this transition and discuss its possible relevance for polymers placed in a random matrix

finite difference reaction diffusion systems with coupled boundary conditions and time delays abstract this paper is concerned with finite difference solutions of a coupled system of reaction diffusion equations with nonlinear boundary conditions and time delays the system is coupled through the reaction functions as well as the boundary conditions and the time delays may appear in both the reaction functions and the boundary functions the reaction diffusion system is discretized by the finite difference method and the investigation is devoted to the finite difference equations for both the time dependent problem and its corresponding steady state problem this investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions monotone convergence of the time dependent solution to a maximal or a minimal steady state solution for quasimonotone functions and local and global attractors of the time dependent system including the convergence of the time dependent solution to a unique steady state solution also discussed are some computational algorithms for numerical solutions of the steady state problem when the reaction function and the boundary function are quasimonotone all the results for the coupled reaction diffusion equations are directly applicable to systems of parabolic ordinary equations and to reaction diffusion systems without time delays

diffusion on regular random fractals we study random walks on structures intermediate to statistical and deterministic fractals called regular random fractals constructed introducing randomness in the distribution of lacunas of sierpinski carpets random walks are simulated on finite stages of these fractals and the scaling properties of the mean square displacement of n step walks are analysed the anomalous diffusion exponents obtained are very near the estimates for the carpets with the same dimension this result motivates a discussion on the influence of some types of lattice irregularity random structure dead ends lacunas on based on results on several fractals we also propose to use these and other regular random fractals as models for real self similar structures and to generalize results for statistical systems on fractals

monotone iterative methods for differential systems with finite delay this paper presents iterative methods for nonlinear differential systems with finite delay the basic idea of the iterative method is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences this approach provides constructive proof of existence as well as numerical procedures for computation of solutions

fully frustrated ising antiferromagnets at the fractal to euclidean crossover abstract we present a study of the residual entropies of the frustrated ising antiferromagnets in a zero field situated on the sierpinski gasket type of fractals our results render it possible to establish the fractal residual entropy crossover towards wannier s exact value for the euclidean triangular lattice

finite difference solutions of reaction diffusion equations with continuous time delays abstract this paper is an extension of the monotone iterative methods for finite difference equations with discrete time delays to a class of nonlinear finite difference system with continuous time delays the system under consideration is a finite difference approximation of a class of reaction diffusion equations with continuous time delays in the nonlinear reaction under either dirichlet or neumann robin boundary conditions various monotone iterative schemes which depend on the property of the nonlinear reaction mechanism are developed for the computation of numerical solutions it is shown by the method of upper and lower solutions that the two sequences obtained from each iterative scheme converge monotonically from above and below respectively to a unique solution of the finite difference system applications are given to two model problems known as the diffusive logistic equation and the fisher s diffusion equation in population genetics

asymptotic form of the spectral dimension at the fractal to lattice crossover the authors study the spectral dimension of the sierpinski gasket family of fractals each member of the family is labelled by an integer b 2 or b infinity so that when b to infinity both the fractal and spectral dimension approach the euclidean value 2 the asymptotic law for the spectral dimension was recently suggested to be ds approximately 2 b 1nb beta with b and beta being some constants here the authors demonstrate that this form should be replaced by the asymptotic law ds approximately 2 1n 1nb constant 1nb their analysis is based on the exact calculation of the electric resistances rb for all members of the family up to b 650

fractal models for diffusion controlled aggregation a two dimensional fractal model is constructed for diffusion controlled deposition on a surface the fractal geometry of the deposit and the power law behaviour of the quantities characterising the non equilibrium cluster size distribution are shown to be consequences of the competition generally present in a nonlinear growth process a qualitative agreement with previous numerical results is found and the scaling laws for the critical exponents of the problem are shown to be satisfied exactly

exact renormalization group treatment of the piecewise directed random walks on fractals abstract an exact evaluation of the critical exponent v associated with the mean squared end to end distance of the piecewise directed random walks is presented in the case of the complete family of the sierpinski gasket type of fractals

studies of the spectral dimension of the vicsek snowflake fractal by an exact real space renormalization group method the authors obtain the spectral dimension of the vicsek snowflake fractal the result seems to cast some doubt on the range of validity of the einstein relation on the deterministic fractal

numerical methods for semilinear parabolic equations the method of upper lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions the idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem the aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable these two schemes are modified jacobi method and gauss seidel method for nonlinear algebraic equations an advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration on the other hand the standard picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations

anisotropy in field assisted diffusion on a gasket fractal the random walk on the two dimensional gasket fractal has been studied with various forms of constant bias fields applied using the monte carlo method crossover from anomalous diffusion to drift was observed in the horizontal and upward bias fields while crossover from anomalous to normal diffusion was found in the downward bias field the scaling relation r bs t approximately tkf bstk was confirmed and the scaling exponent of f x approximately xbeta at large x was obtained as beta 1 175 or 0 004 for the horizontal bias field and beta 0 319 or 0 002 for the downward bias field in good agreement with the theoretical work by parrondo et al 1990

critical behavior of two interacting linear polymer chains in a good solvent a model of two interacting chemically different linear polymer chains is solved exactly using the real space renormalization group transformation on a family of sierpinski gasket type fractals and on a truncated 4 simplex lattice the members of the family of the sierpinski gasket type fractals are characterized by an integer scale factorb which runs from 2 to the hausdorff dimensiond f of these fractals tends to 2 from below asb we calculate the contact exponenty for the transition from the state of segregation to a state in which the two chains are entangled forb 2 5 using arguments based on the finite size scaling theory we show that forb y 2 v b d f wherev is the end toend distance exponent of a chain for a truncated 4 simplex lattice it is shown that the system of two chains either remains in a state in which these chains are intermingled in such a way that they cannot be told apart in the sense that the chemical difference between the polymer chains completely drop out of the thermodynamics of the system or in a state in which they are either zipped or entangled we show the region of existence of these different phases separated by tricritical lines the value of the contact exponenty is calculated at the tricritical points

time delayed finite difference reaction diffusion systems with nonquasimonotone functions this paper is concerned with finite difference solutions of a system of reaction diffusion equations with coupled nonlinear boundary conditions and time delays the reaction functions and the boundary functions are not necessarily quasimonotone and the time delays may appear in the reaction functions as well as in the boundary functions the investigation is devoted to the finite difference system for both the time dependent problem and its corresponding steady state problem some monotone iteration processes for the finite difference systems are given and the asymptotic behavior of the time dependent solution in relation to the steady state solution is discussed the asymptotic behavior result leads to some local and global attractors of the time dependent problem including the convergence of the time dependent solution to a unique steady state solution an application and some numerical results to an enzyme substrate reaction diffusion problem are given all the results are directly applicable to parabolic ordinary systems and to reaction diffusion systems without time delays

dimer coverings on the sierpinski gasket we present the number of dimer coverings nd n on the sierpinski gasket sgd n at stage n with dimension d equal to two three four or five when the number of vertices denoted as v n of the sierpinski gasket is an even number nd n is the number of close packed dimers when the number of vertices is an odd number no close packed configurations are possible and we allow one of the outmost vertices uncovered the entropy of absorption of diatomic molecules per site defined as s mathit sg d lim n to infty ln n d n v n is calculated to be ln 2 3 exactly for sg2 the numbers of dimers on the generalized sierpinski gasket sgd b n with d 2 and b 3 4 5 are also obtained exactly with entropies equal to ln 6 7 ln 28 12 ln 200 18 respectively the number of dimer coverings for sg3 is given by an exact product expression such that its entropy is given by an exact summation expression the upper and lower bounds for the entropy are derived in terms of the results at a certain stage for sgd n with d 3 4 5 as the difference between these bounds converges quickly to zero as the calculated stage increases the numerical value of s mathit sg d with d 3 4 5 can be evaluated with more than a hundred significant figures accurate

critical spin wave dynamics for two dimensional sierpinski type fractal the dynamic critical exponent z for a system of nearest neighbour interacting heisenberg spins on a two dimensional sierpinski type fractal is obtained by applying a real space renormalisation group transformation of the equations of motion

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

critical exponent γ for a class of directed walks at the fractal to euclidean crossover abstract an exact evaluation of the critical exponent associated with the average number of n step walks of the piecewise directed random walks is presented in the case of the complete family of the sierpinski gasket type of fractals

exponentially growing solutions for a delay diffusion equation with negative feedback abstract the scalar partial delay differential equation u t u u t 1 u t is investigated it was known that in one space dimension all nonnegative solutions of the initial boundary value problem stay bounded as t surprisingly this ceases to remain true in two or more dimensions if the delay is large and the diffusion coefficient small there exists a large set of trajectories whose total mass tends exponentially to infinity as t these trajectories escape the negative feedback of the equation by transporting their mass around along periodic paths in space where the paths can be prescribed by the initial data

statistics of equally weighted random paths on a class of self similar structures we study the statistics of equally weighted random walk paths on a family of sierpinski gasket lattices whose members are labelled by an integer b 2 b 2 mean path end to end distance grows more slowly than any power of its length n we provide arguments for the emergence of usual power law critical behaviour in the limit b when fractal lattices become almost compact