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

nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations abstract we consider the dirichlet problem for a class of anisotropic degenerate elliptic equations

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

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

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

positive solutions of inhomogeneous elliptic equations with indefinite data in this paper we study the existence of positive solutions of an inhomogeneous semilinear elliptic equation allowing the inhomogeneous term f x to change sign that is f is an element of c 1 omega 0 we show under further assumptions that the problem has exact one or two positive solutions according to the classification of f c 2004 elsevier ltd all rights reserved

existence results for anisotropic quasilinear parabolic equations with time dependent exponents and gradient term abstract the cauchy dirichlet problem for anisotropic parabolic equation with time dependent exponents and gradient term is considered we state conditions that guarantee the existence and uniqueness of a viscosity solution which is lipschitz continuous in the space variables and holder continuous in time

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

singular and degenerate anisotropic parabolic equations with a nonlinear source abstract the dirichlet problem in arbitrary domain for degenerate and singular anisotropic parabolic equations with a nonlinear source term is considered we state conditions that guarantee the existence and uniqueness of a global weak solution to the problem a similar result is proved for the parabolic p laplace equation

first order phase transitions of the potts model in fractal dimensions abstract the phase diagram of the q state potts model in fractal dimensions is studied with the help of wang landau monte carlo simulations on sierpinski and menger fractal structures a particular attention is paid to first order transitions just above the border separating the second order phase transition regime from the first order one although the translation invariance is strongly broken in deterministic fractals evidence is given that such a deviation from the translational symmetry is not able to induce second order transitions for large values of q when the dimension lies between 1 9746 and 3 moreover the occurrence of second order transitions for very large values of q in the case of hierarchically weakly connected systems that is when the fractal dimension is significantly smaller than 2 is pointed out at last the evolution of first order physical averages such as the latent heats and the interfacial free energies with the space dimensionality and the number of spin states is discussed

on the resistance of the sierpinski carpet let f text n be the nth stage in the construction of the sierpinski carpet let r text n be the electrical resistance of f text n when the left and right sides are each short circuited and a voltage is applied between them we prove that there exists a constant rho such that textstyle frac 1 4 rho n leq r n leq 4 rho n the motivation for this result came from the problem of establishing a the existence and b the value of the spectral dimension of the sierpinski carpet in this and a subsequent paper we settle a and give bounds for b

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

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

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

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

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 phases of antiferromagnetic potts model on fractals the author studies studies the q state antiferromagnetic potts model on fractals namely the sierpinski carpets 1 df 2 and pastry shells 2 df 3 for both families a critical value qc was found above which the system is always in the paramagnetic phase whereas for q at or less than qc the system exhibits a low temperature critical phase the phase diagrams are analyzed from the geometrical point of view

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

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

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

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

a renormalisation group approach to the potts model on the koch curve a three parameter decimation renormalisation is constructed and used to produce exact rg recursion relations for the potts model on the koch curve including all the nearest neighbour interactions only two trivial fixed points are obtained the authors show that no phase transition occurs at finite temperature

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

dirac and magnetic schrodinger operators on fractals abstract in this paper we define local dirac operators and magnetic schrodinger hamiltonians on fractals and prove their essential self adjointness to do so we use the concept of 1 forms and derivations associated with dirichlet forms as introduced by cipriani and sauvageot and further studied by the authors jointly with rockner ionescu and rogers for simplicity our definitions and results are formulated for the sierpinski gasket with its standard self similar energy form we point out how they may be generalized to other spaces such as the classical sierpinski carpet

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

potts model on finitely ramified fractals the author presents an exact solution for the ferromagnetic q state potts model on a family of finitely ramified fractal lattices with fractal dimension between 1 and 2 the author obtains the thermal exponent v for the whole fractal family the results stress the connection between geometrical parameters and critical behaviour opening up new possibilities for modelling physical systems showing fractal structures

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

counting spanning trees on fractal graphs and their asymptotic complexity using the method of spectral decimation and a modified version of kirchhoff s matrix tree theorem a closed form solution to the number of spanning trees on approximating graphs to a fully symmetri

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

self avoiding walks on self similar structures finite versus infinite ramification self avoiding walks saws generated by exact enumeration techniques are studied on the sierpinski carpet d 2 and on the sierpinski sponge d 3 also called sierpinski square lattices a detailed comparison of the results for saws on these infinitely ramified fractals to saws on finitely ramified sierpinski gaskets sierpinski triangular lattices on regular lattices and on the incipient percolation cluster is done providing insight into the behaviour of saws on ordered and disordered structures the saws on sierpinski square lattices are found to display a kind of intermediate behaviour sharing aspects of both saws on ordered and on fractal structures as a consequence a des cloizeaux relation does not seem to hold for this structure as opposed to its validity for saws on regular lattices on sierpinski triangular lattices and on the incipient percolation cluster

bond percolation on branching koch curves with use of the exact recursion equations of the generating functions we determine the asymptotic behaviour of bond percolation on branching koch curves and obtain the critical exponents alpha beta gamma and nu the scaling law 2 alpha gamma 2 beta df nu is obtained this is similar to the scaling law for a regular lattice except that the fractal dimension df is replaced by the euclidean dimension d

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

magnetism of internal surfaces in a fractal structure abstract we study the inhomogeneous magnetization behavior of an ising ferromagnet in sierpinski pastry shells using a real space renormalization group approach two qualitatively different regions on the fractal are distinguished the bulk and the set of internal surfaces which border the eliminated portions we obtain the spontaneous mean magnetizations for these regions as a function of the temperature for different values of js jb js and jb are the internal surface and bulk coupling constants respectively and different geometrical parameters b and l the critical exponents are obtained for the several transitions we obtain different universality classes for the bulk transitions depending on what occurs at the surfaces and a step like behavior of the magnetization as a function of the temperature of some values of b and l

potts model on a fractal lattice the authors consider the dilute potts model on the sierpinski carpet the model has a phase transition at a finite temperature and both critical and tricritical behaviour are observed for qqc where q is the number of potts states they use migdal kadanoff type recursion relations to obtain the value of qc and of critical and tricritical exponents and investigate their dependence on various geometrical characteristics of the fractal

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

absorbing phase transition in conserved lattice gas model on fractal lattices we study the continuous phase transition of the conserved lattice gas clg model from an active phase into an absorbing phase on two fractal lattices i e on a checkerboard fractal and on a sierpinski gasket in the clg model a particle is assumed to be active if any of the neighboring sites are occupied by a particle and inactive if all neighboring sites are empty we estimate critical exponents theta beta nu and nu perpendicular characterizing respectively the density of active particles in time the order parameter the temporal and spatial correlation lengths near the critical point and the results are confirmed by off critical scaling and finite size scaling the order parameter exponent beta on a checkerboard fractal appears to lie between the one dimensional 1d value and two dimensional 2d value of the clg model while that on a sierpinski gasket lies between the 1d and 2d values of the conserved threshold transfer process such a difference is manifested based on the intrinsic properties of the underlying fractal lattices

finite size scaling for random walks on fractals random walks are simulated on finite stages of construction of regular fractal lattices it is proved that the mean square displacement rn2 obeys a finite size scaling hypothesis and the critical exponent nu w is estimated the efficiency of the method is proved when applied to finitely ramified fractals in which the problem is exactly solvable nu w is obtained with good accuracy approximately 1 for a class of infinitely ramified fractals the sierpinski carpets the results correct previous estimates based on simulations which did not use finite size scaling it is shown that nu w decreases when dc decreases with very small corrections due to other geometrical properties such as lacunarity the comparison with estimates of the ideal chain exponent nu c shows that the two problems are not equivalent on these fractals and that in general nu w nu c estimates of nu w with the same accuracy are obtained on two sierpinski pastry shells 2 df 3 where anomalous diffusion is also observed

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

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

characteristic varieties of hypersurface complements abstract we give divisibility results for the global characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface as an application we recast old and obtain new finiteness and divisibility results for the classical infinite cyclic alexander modules of complex hypersurface complements moreover for the special case of hyperplane arrangements we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties

nearby cycles and alexander modules of hypersurface complements abstract let f c n 1 c be a polynomial map which is transversal at infinity using sabbah s specialization complex we give a new description of the alexander modules of the hypersurface complement c n 1 f 1 0 and obtain a general divisibility result for the associated alexander polynomials as a byproduct we prove a conjecture of maxim on the decomposition of the cappell shaneson peripheral complex of the hypersurface moreover as an application we use nearby cycles to recover the mixed hodge structure on the torsion alexander modules as defined by dimca and libgober we also explore the relation between the generic fibre of f and the nearby cycles

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

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

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

q plane zeros of the potts partition function on diamond hierarchical graphs we report exact results concerning the zeros of the partition function of the potts model in the complex q plane as a function of a temperature like boltzmann variable v for the m th iterate graphs dm of the diamond hierarchical lattice including the limit m in this limit we denote the continuous accumulation locus of zeros in the q planes at fixed v v0 as bq v0 we apply theorems from complex dynamics to establish the properties of bq v0 for v 1 the zero temperature potts antiferromagnet or equivalently chromatic polynomial we prove that bq 1 crosses the real q axis at i a minimal point q 0 ii a maximal point q 3 iii q 32 27 iv a cubic root that we give with the value q q1 1 638 896 9 and v an infinite number of points smaller than q1 converging to 32 27 from above similar results hold for bq v0 for any 1 0 potts ferromagnet we also provide the computer generated plots of bq v0 at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of z d4 q v0

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

inhomogeneity and complexity measures for spatial patterns abstract in this work we examine two different measures for inhomogeneity and complexity that are derived from non extensive considerations a la tsallis their performance is then tested on theoretically generated patterns all measures are found to exhibit a most sensitive behaviour for sierpinski carpets the procedures here introduced provide us with new powerful tsallis tools for analysing the inhomogeneity and complexity of spatial patterns

frustration and correlations in fractals abstract a technique for obtaining exact thermodynamic and correlation functions of regular fractals is illustrated by a development for the ising system on the triangular sierpinsky gasket a generalised hyperscaling statement is obtained the technique is then applied to the fully frustrated antiferromagnetic case where the exact ground state entropy per spin is shown to be s 0 0 4930 this result shows that the introduction of regularly disposed holes into the triangular lattice does not relieve frustration as is commonly supposed relations completely determining the correlation functions are obtained that for the zero field pair correlation function of vertex spins at separation r is g 1 2 r g 2 r g 1 together with a relation which sets the correlation length scale exact asymptotic expressions are derived for ferromagnetic correlations in both zero and non zero field the zero field zero temperature correlation length of the fully frustrated system is shown to be 0 79 representing extremely rapid decay of correlations

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

vecteurs distributions h invariants pour les series principales generalisees d espaces symetriques reductifs et prolongement meromorphe d integrales d eisenstein

critical behaviour of the long range interaction model on a fractal lattice a particular class of fractal lattices is considered where the ising model with power law decaying interactions can be well approximated to a hierarchical type model applying the exact recurrence relations the critical behaviour is calculated and its universality with respect to the fractal dimensionality d is examined

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

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

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

eden trees on the sierpinski gasket the authors have studied eden trees on the sierpinski gasket and obtained its fractal and spectral dimensions to be ln 7 square root 5 2 ln 2 and 1 respectively

the problem of dirichlet for anisotropic quasilinear degenerate elliptic equations we consider the dirichlet problem for a class of anisotropic degenerate elliptic equations new a priori estimates for solutions and for the gradient of solutions are established based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated the results are new even in the semilinear case when the principal part is the laplace operator

existence of lipschitz continuous solutions to the cauchy dirichlet problem for anisotropic parabolic equations abstract the cauchy dirichlet and the cauchy problem for the degenerate and singular quasilinear anisotropic parabolic equations are considered we show that the time derivative u t of a solution u belongs to l under a suitable assumption on the smoothness of the initial data moreover if the domain satisfies some additional geometric restrictions then the spatial derivatives u x i belong to l as well in the singular case we show that the second derivatives u x i x j of a solution of the cauchy problem belong to l 2

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

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

ising model on a 3d sierpinski gasket with a non trivial phase transition the author derives and solves exact recursion relations for a multispin interaction ising model on a 3d sierpinski gasket in addition to the high and low temperature fixed points he finds three non trivial fixed points which govern the behaviour of two new multiple of 2 phases these phases are characterised by the absence of any tetrahedron with an odd number of up spins

magnetization of a potts ferromagnet on a sierpinski carpet the authors study the inhomogeneous magnetization behaviour of a q state potts ferromagnet on a sierpinski carpet as a function of the temperature using a real space renormalization group procedure two qualitatively different regions on the carpet are distinguished a the interior of the carpet and b the set of internal walls which border the eliminated areas the spontaneous mean magnetization curves for these regions are obtained with the corresponding critical exponents beta and beta w for different values of q they find that the magnetization of region b goes to zero at the critical temperature always below the one of region a

vibration modes of 3n gaskets and other fractals we rigorously study eigenvalues and eigenfunctions vibration modes on the class of self similar symmetric finitely ramified fractals which include the sierpinski gasket and other 3n gaskets we consider the classical laplacian on fractals which generalizes the usual one dimensional second derivative is the generator of the self similar diffusion process and has possible applications as the quantum hamiltonian we develop a theoretical matrix analysis including analysis of singularities which allows us to compute eigenvalues eigenfunctions and their multiplicities exactly we support our theoretical analysis by symbolic and numerical computations our analysis in particular allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues i e integrated density of states we consider such examples as the level 3 sierpinski gasket a fractal 3 tree and the diamond fractal

local systems on analytic germ complements abstract we prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori this is a generalization of the classical monodromy theorem stating that the eigenvalues of the monodromy on the cohomology of the milnor fiber of a germ of a holomorphic function are roots of unity