geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices it is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal lattices of low lacunarity these lattices are special examples of sierpinski carpets their being of low lacunarity means that they are arbitrarily close to being translationally invariant the claim is substantiated for the ising model in d 1 ensuremath epsilon dimensions and for resistor network models with 1ldl2

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

critical exponents of the ising model on low dimensional fractal media abstract the critical behavior of the ising model on fractal substrates with noninteger hausdorff dimension d h 2 and infinite ramification order is studied by means of the short time critical dynamic scaling approach our determinations of the critical temperatures and critical exponents and are compared to the predictions of the wilson fisher expansion the wallace zia expansion the transfer matrix method and more recent monte carlo simulations using finite size scaling analysis we also determined the effective dimension d e f which plays the role of the euclidean dimension in the formulation of the dynamic scaling and in the hyperscaling relationship d e f 2 furthermore we obtained the dynamic exponent z of the nonequilibrium correlation length and the exponent that governs the initial increase of the magnetization our results are consistent with the convergence of the lower critical dimension towards d 1 for fractal substrates and suggest that the hausdorff dimension may be different from the effective dimension

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

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

scaling exponents for random walks on sierpinski carpets and number of distinct sites visited a new algorithm for infinite fractal lattices the scaling exponent for the mean square distance covered in a random walk dw and the average number of distinct sites visited dn are determined for a family of sierpinski carpet patterns we suggest a new random walk algorithm to generate walks on an effectively infinite deterministic fractal lattice the algorithm is applied to several sierpinski carpet patterns with the same hausdorff dimension we show that the systems have a quite different scaling exponent dw and further that the generally accepted result dn ds does not hold for all of these where ds is the spectral dimension

hopf bifurcation and stability in predator prey model with a stage structure for prey a predator prey system with b d functional response and stage structure for prey is presented in the d t plane the regions of stability and hopf bifurcation are found by analyzing the associated characteristic transcendental equation further global stability of equilibria are studied by using comparison theorem some numerical simulations are also given to illustrate our results

coexistence and stability of predator prey model with beddington deangelis functional response and stage structure abstract we present a predator prey model of beddington deangelis type functional response with stage structure on prey the constant time delay is the time taken from birth to maturity about the prey by the uniform persistence theories and monotone dynamic theories sharp threshold conditions which are both necessary and sufficient for the permanence and extinction of the model as well as the sufficient conditions for the global stability of the coexistence equilibria are obtained biologically it is proved that the variation of prey stage structure can affect the permanence of the system and drive the predator into extinction by changing the prey carrying capacity our results suggest that the predator coexists with prey permanently if and only if predator s recruitment rate at the peak of prey abundance is larger than its death rate and that the predator goes extinct if and only if predator s possible highest recruitment rate is less than or equal to its death rate furthermore our results also show that a sufficiently large mutual interference by predators can stabilize the system

lacunarity and universality lacunarity as a criterion of universality in phase transitions on sierpinski carpets is studied the approximate theoretical and numerical results suggest that systems of different lacunarity may belong to a common universality class there seems to be a lack of solutions to the complete set of universality criteria on sierpinski carpets

potts model on sierpinski carpets and their relation to hypercubic lattices of non integer dimensionality the authors study the critical frontiers and the high and low temperature asymptotic behavior of the potts model two site correlation function for several sierpinski carpets with fractal dimension lying between one and two they simulate the sierpinski carpets by means of appropriate hierarchical lattices obtaining from their basic cells the two graphs among others that arise from the anisotropic bond moving schemes they correspond to the asymptotic high and low temperature dominant terms for the hierarchical lattice two site correlation function exact result by taking the infinite limit size of the basic cell these asymptotic behaviors tends to those of the sierpinski carpets it is then possible to compare them with similar results obtained from analytical extension to d 2 of the hypercubic lattice their results indicate that these asymptotic behaviors do not always coincide suggesting that possibly they may not identify the sierpinski carpets exhibited by gefen et al 1984 as a correct implementation of the analytical extension of hypercubic lattices

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

a mathematical construction of the non abelian chern simons functional integral we construct rigorously an infinite dimensional distribution which corresponds to the chern simons cs functional integral associated with a principal fiber bundle over r 3 with structure group a compact connected lie group we determine the moments of the cs distribution and show that these coincide with those used in informal studies of the cs integral a locality property of the cs distribution is proven the complexified theory of frohlich and king is also discussed within our framework

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

asymptotic behaviors of competitive lotka volterra system with stage structure in this paper a two species competitive model with stage structure is presented and studied results on the global extinction and permanence are given which generalize the well known three theorems for the two species competitive system and moreover they confirm the negative effect of stage structure on the permanence of populations as well as estimate the degree of such effect conclusions in this paper suggest that for a competitive community stage structure is also one of the important reasons that cause permanence and extinction

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

exact renormalization group approach to critical phenomena in an ising system with long range interactions on the sierpinski gasket lattice abstract an ising spin system with two point long range interactions on the sierpinski gasket lattice is considered the interactions are assigned along edges of all equilateral triangles being part of the self similar structure in addition the long range couplings i e all couplings besides nearest neighbor interactions k are assumed to be independent of the distance i e q1 q l 1 2 where q1 denotes an interaction between the two spins on the distance 2 a with a being a lattice constant and q a positive parameter using an exact renormalization group method based upon the decimation transformation the system is shown to display a line of nontrivial fixed points k k q each point of this line corresponds to a phase transition of a rather high order the critical behavior of the system near every nontrivial fixed point is claimed to be characterized by two dimensions viz the fractal dimension and the so called critical dimension which reflects some topological properties of the system the critical dimension is defined through considering the decay of an effective two point correlation function and depends on q

critical behaviour of the bond diluted potts model on sierpinski carpets a four parameter bond moving renormalisation is used to study the critical properties of the bond diluted q state potts model on sierpinski carpets fixed points critical exponents and a phase diagram are obtained in the authors results there is a borderline value q qt for q qt there is crossover to a new diluted fixed point this behaviour is similar to that of the diluted system on a regular lattice with translational invariance they give some values of qt for carpets with different b l where b and l are structure parameters of carpets

critical behavior of self avoiding walks on fractals using a new graph counting technique suitable for self similar fractals exact 18th order series expansions for saws on some sierpinski carpets are generated from them the critical fugacityxc and critical exponents saw and saw are obtained the results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied we find for some carpets with low lacunarity that saw saw d for saws on the fractals which are embedded in ad dimensional euclidean space

global analysis of the predator prey system with beddington deangelis functional response abstract in this paper predator prey systems with beddington deangelis functional response are considered by using divergency criterion global stability is established provided the system possesses a positive locally asymptotically stable equilibrium

uniqueness of limit cycles of the predator prey system with beddington deangelis functional response abstract the goal of this paper is to establish the uniqueness of limit cycles of the predator prey systems with beddington deangelis functional response through a change of variables the predator prey system can be transformed into a better studied gause type predator prey system as a result the uniqueness of limit cycles can be solved

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

on the dynamics of predator prey models with the beddington deangelis functional response abstract the dynamics of predator prey models with the beddington deangelis functional response are analyzed primarily from the viewpoint of permanence uniform persistence the beddington deangelis functional response is similar to the holling type 2 functional response but contains an extra term describing mutual interference by predators both spatially homogeneous models based on ordinary differential equations and reaction diffusion models are considered criteria for permanence and for predator extinction are derived for systems without diffusion or with no flux boundary conditions criteria are derived for the existence of a globally stable coexistence equilibrium or alternatively for the existence of periodic orbits

the ideal chain problem in infinitely ramified self similar structures abstract series expansions for the ideal chain problem in sierpinski carpets were calculated and critical exponents 1 2 were obtained with good accuracy from the scaling properties of the probability of the chain returning to the starting site it is shown that the ideal chain has asymptotic behaviour different from the random walk problem in those lattices

the chern simons theory and knot polynomials the chern simons gauge theory is studied using a functional integral quantization this leads to a differential equation for expectations of wilson lines the solution of this differential equation is shown to be simply related to the two variable jones polynomial of the corresponding link in the case where the gauge group issu n a similar equation has been used before to get the jones polynomial from a braid representation of the link the main novelty of our approach is that we get the jones polynomial from a plat representation of the link

critical properties of the s4 model on diamond type hierarchial lattices we study the critical properties of the s 4 model on diamond type hierarchical lattices in the presence of an external magnetic field it is assumed that for this type of inhomogenous fractal lattice the gaussian distribution constant the four spins interaction parameter and the external magnetic field in the s4 model depend on the coordination number of the site on the fractal lattices by combining the real space renormalization group scheme with the cumulative expansion method we obtain the critical points and further calculate critical exponents according to the scaling theory the results show that on diamond type hierarchical lattices with branches m 4 of 2 bonds the critical point of the s4 model is just the gaussian fixed point and therefore critical exponents are in full agreement with those of the gaussian model and that on those with m 4 the wilson fisher fixed point as well as the gaussian fixed point is obtained the wilson fisher fixed point has a decisive influence on the critical behavior of the system and critical exponents are related to the fractal dimensionality we found that the s4 model on the fractal lattices and that on the translation symmetric lattices show similar behaviors in the dependence of the critical properties on the dimensionality c 2005 elsevier b v all rights reserved

application of bond moving renormalization group approach to fractal lattices we investigate the application of the migdal kadanoff bond moving renormalization group rg approach to fractal lattices we find the following two results first for inhomogeneous interaction lattice models bond moving involving inequivalent bonds is unsuitable because it violates the condition 0 is the perturbation potential resulting from moving the bonds second the condition 0 does not uniquely determine the way to move bonds different choices of bond moving yield different rg recursion relations and corresponding fixed points which makes the conclusions concerning the phase transition quite uncertain

resistance and spectral dimension of sierpinski carpets the authors announce results which prove the existence of the spectral dimension ds for sierpinski carpets in two dimensions their method employs the einstein relation zeta d omega df using this numerical calculations of the resistance of approximations to the sierpinski carpet yield an accurate estimate for ds

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

dilute ising model on fractal lattices the authors study dilute ising models on a family of infinitely ramified exact fractals whose fractal dimension varies from 1 to 2 they calculate the phase diagram and critical exponents as a function of the fractal dimensionality within migdal s approximation 1975

fractals their transfer matrices and their eigen dimensional sequences this paper defines and studies a general algorithm for constructing new families of fractals in euclidean space this algorithm involves a sequence of linear interscale transformations that proceed from large to small scales the authors find that the fractals obtained in this fashion decompose in intrinsic fashion into linear combinations of a variable number of addend fractals the addends relative weights and fractal dimensionalities are obtained explicitly through an interscale matrix which they call the transfer matrix of the fractal tmf the authors first demonstrate by a series of examples then prove rigorously that the eigenvalues of our tmfs are real and positive and that the fractal dimensions of the addend fractals are the logarithms of the eigenvalues of their tmf they say that these dimensions form the overall fractal s eigendimensional sequence the eigenvalues of their tmf are integers in the non random variants of the construction but are non integer in the random variants a geometrical interpretation of the eigenvalues and the eigenvectors is given their tmf have other striking and very special properties that deserve additional attention

knot theory and the heuristics of functional integration abstract this paper is an exposition of the relationship between the heuristics of witten s functional integral and the theory of knots and links in three dimensional space

studies of the spectral dimension for branched koch curves the scaling behaviour of koch curves with a rather general branched structure is studied by means of the exact renormalisation method and the general formula of spectral dimension is obtained it is proved that the spectral dimension is only dependent on the topology of the curve the einstein relation between the random walk dimension fractal dimension and size scaling exponent of dc conductivity is also checked

lower and upper bounds for the anomalous diffusion exponent on sierpinski carpets the anomalous diffusion exponent dw of random walks on a family of sierpinski carpets are studied by analytical and numerical methods we construct an effective bulk resistor and then establish the lower and upper bounds for dw where the lower bound turns out to be the same as that obtained with bond moving renormalization numerical simulations on a family of sierpinski carpets confirm our bounds and show strong dependence of dw on the lacunarity of the carpet

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

critical properties of ising models with long range interactions on sierpinski gasket lattices in higher euclidean dimensions abstract ising systems with two spin long range interactions are studied on sierpinski gasket lattices generalized to euclidean dimensions d 3 4 5 6 for each lattice the long range interactions are assigned to edges that are part of the appropriate self similar structure in consequence these systems are infinitely ramified and exhibit critical behavior at finite temperatures an analysis of critical properties of the systems in particular the calculation of critical exponents is carried out by using an exact renormalization group method it is shown that there exists some similarity between critical properties of the long range interacting fractal models under study here and critical properties of short range interacting ising models defined on abstract regular lattices interpolated to noninteger dimensionalities

lacunarity and universality it is shown that the new expression for the lacunarity of sierpinski carpets proposed by taguchi 1987 is not appropriate and that the new expression of lacunarity fractal dimension and connectivity together do not specify uniquely critical exponents of ising models with spins on the carpet

lacunarity and percolation phase transitions the lacunarity is computed for randomly diluted two dimensional lattices in terms of the chemical distance the crossover behaviour and the question of universality are discussed

a suggested generalization for the lacunarity index abstract nowadays lacunarity is a well accepted concept to complement fractality studies of complex sets lacunarity features are used to assess the largeness of gaps or holes of complex signals and images as it stands the lacunarity analysis used is an index averaging the behavior of small and large size structures in a given set the aim of this work is to propose a generalized lacunarity concept on the basis of generalized moments in this form the lacunarity index depending on a discrimination parameter is oriented to quantify lacunarity associated with objects of different size textbook and real images are used to illustrate the ability of the generalized lacunarity index for displaying interesting features of real complex sets

critical behavior of the gaussian model on a diamond type hierarchical lattice with periodic and aperiodic interactions abstract the critical behavior of the gaussian models on a diamond type hierarchical lattice with periodic and aperiodic interactions according to a class of substitution sequences are studied by the exact renormalization group approach and spin rescaling for the periodic case with a two letter general fibonacci sequence given by the generating substitutions a ab b ab and the aperiodic case with the generating substitutions a ab b ba the critical behaviors are found unchanged with respect to the uniform case and belong to the same universality of the uniform case for the other aperiodic case with the generating substitutions a ab b aa the geometric fluctuation is relevant by luck s criterion and the critical phenomena of the uniform and periodic cases are found to disappear

dynamics of a nonautonomous predator prey system with the beddington deangelis functional response abstract in this paper we systematically study the dynamics of a nonautonomous predator prey system with the beddington deangelis functional response the explorations involve the permanence extinction global asymptotic stability general nonautonomous case the existence uniqueness and stability of a positive almost periodic solution and a boundary almost periodic solution for the periodic almost periodic case the paper ends with some interesting numerical simulations that complement our analytical findings

almost periodic solution for a volterra model with mutual interference and beddington deangelis functional response in this paper we consider a volterra model with mutual interference and beddington deangelis functional response by applying the comparison theorem of the differential equation and constructing a suitable lyapunov functional sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained

a stage structured predator prey model of beddington deangelis type we formulate and study a robust stage structured predator prey model of beddington deangelis type functional response the time delay is the time taken from birth to maturity the beddington deangelis functional response is similar to the holling type 2 functional response but contains an extra term describing mutual interference by predators first we show that the predator coexists with prey permanently if and only if the predator s recruitment rate at the peak of prey abundance is larger than its death rate second we show that if the system is permanent then a sufficiently large degree of the predator interference can not only stabilize the system but also guarantee the stability of the system against the increase of the carrying capacity of prey and the increase of birth rate of the adult predator third we show both analytically and numerically that stability switches of interior equilibrium may occur as maturation time delay increases stability may change from stable to unstable to finally stable implying that a large delay can be stabilizing

the coastline and lake shores on a fractal island we compute the fractal dimensions of the hulls or external boundary and the boundaries of the internal cavities in several deterministic as well as random fractal structures our conclusion is that the two fractal dimensions are in fact identical the deterministic fractals we study are sierpinski carpets sc in a two dimensional space and the random fractals are percolation clusters at criticality as an intermediate case we present results on some randomized sc in the random structures statistics of the area and perimeters of all internal cavities or holes are taken and the fractal dimension of the hull borderline is computed two different definitions of the borderline are used considering nearest neighbours as well as nearest and second nearest neighbours as connected the conclusion is valid for both cases

hysteresis scaling for ising systems on fractal structures dynamical phase transitions in ising systems on sierpinski carpets and bond percolation lattices at percolation threshold are studied by means of standard monte carlo simulations we find that the area of hysteresis loop a can be scaled with respect to the sweep rate h of a linear driving field however the exponent in the scaling expression a hb is universal only for ising systems on sierpinski carpets we conclude that the hysteresis scaling is universal for the field driven first order phase transitions in ising systems on fractal structures based on scaling hypothesis we derive the expression of finite size effect on the hysteresis the exponent b is obtained by this method in some sierpinski carpets

noninteger dimensional hyper euclidean lattices on sierpinski carpets random walks on sierpinski carpets are studied with bond moving renormalisation using the results the author constructs fractal lattices whose fractal dimension of random walk is two such fractal lattices are called hyper euclidean lattices it turns out that hyper euclidean lattices have a connectivity which is one smaller than the fractal dimension

functional integration and the theory of knots this paper examines the role of functional integration in the theory of knots and links it is shown how it is possible to find many properties and conjectures about link invariants via this point of view and how these considerations lead to ideas about the mathematical status of such forms of integration

direct evidence for weak universality on fractal structures abstract we provide a direct quantitative evidence that the critical behavior of the ising model on self similar lattices depends upon the geometrical features of the underlying structure intensive monte carlo simulations enable to show that the values of the ratio of exponents depend upon the lacunarity of the structure which measures the deviation from the translational symmetry finally the topological character of the scaling corrections when the fractal dimension is smaller than 2 is discussed

a suggested lacunarity expression for sierpinski carpets a suggested approximate expression for lacunarity is proposed some numerical investigations show that the old expression fails in describing the homogeneity of elimination for some sierpinski carpets

stability and bifurcation in a delayed predator prey system with beddington deangelis functional response we consider a delayed predator prey system with beddington deangelis functional response the stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation by choosing the delay as a bifurcation parameter we show that hopf bifurcation can occur as the delay crosses some critical values the direction and stability of the hopf bifurcation are investigated by following the procedure of deriving normal form given by faria and magalhaes an example is given and numerical simulations are performed to illustrate the obtained results

vassiliev knot invariants and chern simons perturbation theory to all orders at any order the perturbative expansion of the expectation values of wilson lines in chern simons theory gives certain integral expressions we show that they all lead to knot invariants moreover these are finite type invariants whose order coincides with the order in the perturbative expansion together they combine to give a universal vassiliev invariant

lacunarity and universality a new expression for the lacunarity of sierpinski carpets is proposed it turns out that the universality can be described with the new expression the fractal dimension and the connectivity

scale dependent lacunarity and fractal ising models as an alternate way of measuring the lacunarity of sierpinski carpets the authors have computed the site to site fluctuation of the quantity m n namely the total number of lattice sites within the chemical distance of n in long scales of n the resultant lacunarity agrees qualitatively with the results obtained in the past using the covering method in short scales of n however the lacunarity shows a different trend among fractals it varies monotonically with the ising correlation length exponent a trend which was sought before but not found with the previous method this applies only for the carpets of the first kind

change in order of phase transitions on fractal lattices abstract we re examine a population model which exhibits a continuous absorbing phase transition belonging to directed percolation in 1d and a first order transition in 2d and above studying the model on sierpinski carpets of varying fractal dimensions we examine at what fractal dimension 1 d f 2 the change in order occurs as well as commenting on the order of the transitions we produce estimates for the critical points and for continuous transitions some critical exponents

phase transitions in regularly diluted 2d ising models with a complicated inherent structure of a lattice cell abstract we analyze the critical point and study the nonuniversal peculiarities occuring in the temperature dependence of the specific heat for a sequence of the exactly solvable highly decorated two dimensional triangular type ferromagnetic ising models with a complicated fractal type inherent structure of elementary cell the lattices are indexed by the decoration parameter n 0 1 2 3 the elementary cell for the n lattice is the n th order lattice approximation of the sierpinski gasket fractal at any finite n there is the second order phase transition indicated by the logarithmic singularity in the specific heat the critical temperature decreases as 4 ln 4 n 1 with increasing decoration the specific heat critical amplitude vanishes as 3 n n ln n 2 as n the basic nonuniversal features observable in the specific heat at strong decoration are the anomalous narrowing of the critical interval and the appearance of a rounded peak above the true t c point these effects are closely related and can both be explained by the above t c local ordering in the local lattice inhomogeneities the mechanism of the above t c local ordering is analyzed the models under discussion can as well be interpreted as the regular bond diluted 2d ising ferromagnets with a hierarchy of the nonmagnetic holes at different scales and varying dilution a comparative discussion on the properties of the regularly and randomly diluted 2d ising ferromagnets is given

exact results for an ising system with long range interactions on the sierpinski gasket lattice thermodynamic properties of an ising model with two point long range interactions on the sierpinski gasket lattice are studied using an exact recursive methodfor calculating the partition function the long range interactions are introduced in a self similar manner reflecting the structure of the lattice and are assumed to be of the form ql q 2 l 1 where q 0 0 are parameters and l 1 2 indicates that ql couples pairs of spins on the distance 2l measured in units of the lattice spacing in the case of constant long range interactions 0 exact analytic expression for the critical temperature and the critical exponent v are derived it is also shown that the recursive method can be applied to analyze the problem of the existence of phase transitions at finite temperatures in the case of 0 i e when the long range coupling decrease with the distance

self similar and translationally invariant structures in a bond diluted ising model a bond diluted ising model is used to simulate a process from a cantor set to a translationally invariant lattice the free energy function shows that there is no singularity which seem to imply that no transition exists in this process

perturbative expansion of chern simons theory with non compact gauge group naive imitation of the usual formulas for compact gauge group in quantizing three dimensional chern simons gauge theory with non compact gauge group leads to formulas that are wrong or unilluminating in this paper an appropriate modification is described which puts the perturbative expansion in a standard manifestly unitary format the one loop contributions which differ from naive extrapolation from the case of compact gauge group are computed and their topological invariance is verified

lacunarity calculation in the true fractal limit we measure the lacunarity of regular fractals in the limit of an infinite fractal lattice using a suitable graph counting method the calculations are performed for a recent expression proposed for lacunarity that explores the scaling properties of the mass distribution of the fractal set on cells of radius r we present results for various sierpinski carpets and for the sierpinski gasket

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

renormalization group approach to the multi component site percolation on sierpinski carpets abstract the multi component site percolation problems on a family of sierpinski carpets scs are solved by the real space renormalization group rg method three sorts of sites on an sc lattice are distinguished and three independent kinds of occupation probabilities are assigned to them we develope the usual choice of a cell on the translationally invariant lattices and choose suitable cells to construct the rg transformation scheme based on the connectivity the non trivial percolation threshold values and correlation length exponents on such fractals are obtained

dynamics of a density dependent stage structured predator prey system with beddington deangelis functional response abstract in this paper we study the dynamics of a stage structured predator prey system with beddington deangelis functional response where not only the prey density dependence but also the predator density dependence are considered such that the studied predator prey system conforms to the realistically biological environment first we derive a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of hyperbolic curves then we provide a sufficient condition to assure the local asymptotic stability of this positive equilibrium by constructing a lyapunov function afterward by iteratively making use of the comparison theorem we propose a sufficient condition to assure its global attractiveness finally by investigating types of the limit set instead of making use of the persistence theory we prove that the predator coexists with the prey permanently if and only if positive equilibria exist

classification and universal properties of sierpinski carpets the type of critical points for the potts model is found to be dependent on the structure of sierpinski carpets all possible types of structure for sierpinski carpets with two interactions after bond moving are found the author presents general sierpinski carpets and new parameters are proposed and used to describe them the lacunarity expression is revised the author discusses and proposes a universal classification critical points eigenvalues and flow diagrams are presented some fixed points display negative eigenvalues

potts model on sierpinski carpets migdal kadanoff bond moving renormalisation is used to study the q state potts model on sierpinski carpets a general approximate recursion relation including q as a parameter is given the property of bond interchanging invariance is found and used in deriving the recursion relation fixed points and critical exponents for some carpets are presented marginal fixed points instead of unstable ones are found several typical flow diagrams are also shown

extinction and permanence in nonautonomous competitive system with stage structure in this paper the n species nonautonomous stage structured competitive system is constructed and considered sufficient conditions for its extinction and permanence are obtained results here generalize and unify some previous ones moreover it is concluded that stage structure in this system is one of the important factors that effect the extinction and permanence of species

global asymptotic stability of a stochastic delayed predator prey model with beddington deangelis functional response this paper is concerned with a stochastic predator prey system with beddington deangelis functional response and time delay sufficient conditions for global asymptotic stability of the positive equilibrium point are established the result demonstrates that the delay is harmless for global stability of the stochastic system in some cases some figures are given to validate the analytical results

kontsevich integral for vassiliev invariants from chern simons perturbation theory in the light cone gauge we analyze the structure of the perturbative series expansion of chern simons gauge theory in the light cone gauge after introducing a regularization prescription that entails the consideration of framed knots we present the general form of the vacuum expectation value of a wilson loop the resulting expression turns out to give the same framing dependence as the one obtained using nonperturbative methods and perturbative methods in covariant gauges it also contains the kontsevich integral for vassiliev invariants of framed knots

knot polynomials and vassiliev s invariants a fundamental relationship is established between jones knot invariants and vassiliev s knot invariants since vassiliev s knot invariants have a firm grounding in classical topology one obtains as a result a first step in understanding the jones polynomial by topological methods

absence of phase transition in the xy model on menger sponge abstract we have performed a monte carlo study of the classical xy model on a menger sponge with the wolff cluster algorithm u wolff 1989 the menger sponge is a fractal object with infinite order of ramification and fractal dimension d log 20 log 3 2 7268 from the dependence of the helicity modulus on system size and on boundary conditions we conclude that there is no phase transition in the system at any finite temperature

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible