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

effects of random and deterministic discrete scale invariance on the critical behavior of the potts model the effects of disorder on the critical behavior of the q state potts model in noninteger dimensions are studied by comparison of deterministic and random fractals sharing the same dimensions in the framework of a discrete scale invariance we carried out intensive monte carlo simulations in the case of a fractal dimension slightly smaller than two d f 1 974636 we give evidence that the disorder structured by discrete scale invariance does not change the first order transition associated with the deterministic case when q 7 furthermore the study of the high value q 14 shows that the transition is a second order one both for deterministic and random scale invariance but that their behavior belongs to different university classes

order propagation in dilute antiferromagnetic potts models the possible orderings in a dilute antiferromagnetic q 3 potts model on a triangular lattice are studied by applying to a hierarchical lattice an approximate renormalisation group scheme which incorporates and preserves all the symmetries of the potts model this scheme indicates that states with partial order in which only a subset of the symmetries are broken can exist in systems with these symmetries they do not however occur in the model with nearest neighbour pair interactions only in this model the authors find solely the antiferromagnetically ordered states which occur for bond probabilities in excess of p1 which is greater than pc the critical probability for bond percolation

time dependent ginzburg landau model in the absence of translational invariance non conserved order parameter domain growth we have determined the static and dynamical properties of the ginzburg landau model with global coupling of the spherical type on some non translationally invariant lattices our solutions show that in agreement with general theorems fractal lattices with finite ramification do not display a finite temperature phase transition for any embedding dimension d on the other hand the dynamical behaviour associated with the phase ordering dynamics of a non conserved order parameter is non trivial our analysis reveals that the domain size r grows in time as and relates this exponent to the three exponents which characterize the static and dynamical properties of fractal structures namely the fractal dimension of the lattice the random walk dimension and the spectral dimension we also present a brief renormalization group treatment of the model finally we have considered lattices with infinite ramification numbers which have spectral dimensions larger that 2 and show a finite temperature phase transition

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

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

on linear and exponential type non linear lattices a method of linearisation of the infinite toda lattice is proposed which enables one to recognise the relation between the infinite toda lattice and the harmonic lattice straightforwardly

representation theorems for t wright convexity abstract in the present paper we prove some representation theorems for t wright convex functions as a consequence of a support theorem which was proved by the author in earlier paper

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

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

an alternative expression for the addition theorems of spherical wave solutions of the helmholtz equation an alternative formulation of the addition theorem for spherical wave solutions of the helmholtz equation is presented the 3 j symbols of wigner or more precisely the gaunt coefficient complete solid angle integral of a triple product of spherical harmonics which appear in the formerly introduced expressions of these addition theorems are replaced by an explicit matrix expression relating the spherical wave solutions defined with respect to the different origins the generalized gaunt coefficients which are complete solid angle integrals of a multiple product of spherical harmonics can then be written in terms of a matrix product of basic matrices representing the gaunt coefficient

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

directed percolation a finite size renormalisation group approach the finite size renormalisation group technique introduced by nightingale 1976 is applied to the directed percolation problem the decay of correlations is anisotropic in this model and finite size scaling is extended to treat such anisotropy precise estimates for critical exponents and percolation probabilities are obtained for site bond and site bond percolation on the square lattice with bonds directed along the positive axes both free boundary conditions for which the results converge linearly with 1 n as n to infinity and helical boundary conditions for which unexpectedly the results converge linearly with 1 n3 are considered

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

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

a hierarchical percolation model some exact results on the sierpinski gasket a hierarchical percolation model is formulated on the sierpinski gasket with the use of position space renormalisation group methods exact results are obtained that place this model in a distinct universality class from that of the geometric percolation model for the first of two cases considered pc 1 nu 1 d and for the second pc 1 2 nu 1 d 1 where d is the fractal dimension of the sierpinski gasket the hierarchical model is also formulated on the triangular lattice and some preliminary results are obtained

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

nonlocal percolation and partial ordering in the dilute baxter wu model the zero temperature limit of the dilute baxter wu model is studied in the context of percolation theory the percolation model describing the propagation of order is identified and shown to possess a nonlocal mechanism for the propagation of that order a monte carlo study of this model is performed using the nonlocal cluster identification algorithm developed by fried and schick 1992 the author also shows that the mechanism responsible for nonlocal percolation allows for the possibility of the propagation of partial order at intermediate impurity concentration a partially ordered phase is found and its structure is discussed

addition theorems for spherical wave solutions of the vector helmholtz equation addition theorems for spherical wave solutions of the vector helmholtz equation are discussed the theorems allow one to expand a vector spherical wave about a given origin into spherical waves about a shifted origin a simplified derivation of the results obtained earlier by cruzan o r cruzan q appl math 20 33 1962 is presented

representation theorems for h convexity the present paper deals with h convex functions introduced by s varosanec in 21 in our main result we give a characterization of h convex functions satisfying the condition h h 1 1 0 1

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

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

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

diffusion percolation i infinite time limit and bootstrap percolation several models for the dynamic growth of percolation clusters or diffusion percolation dp are introduced and analysed in these models a random walker an ant walks on percolation clusters which are occupied with initial site concentration pi the ant is allowed to step off such clusters and add new sites to them if certain conditions are met some of these models are shown to have a one to one correspondence with models of bootstrap percolation bp in which sites which do not have a required number of neighbours are successively culled two new percolation thresholds have been calculated for two diffusion percolation models on the square lattice

three state potts antiferromagnet on the triangular lattice the three state potts antiferromagnetic model on the triangular lattice 3paft has a weak first order transition in the pure case the first order nature follows from analytic arguments and while it has been observed in some simulations the precision of and internal disagreement between these observations has left much to be desired the 3paft is the only two dimensional nearest neighbour potts model with an ordered phase that does not have an infinite degeneracy that remains without an exact value for its critical point this model is expected to exhibit interesting glassy behaviour and a possible crossover to a second order transition upon dilution a careful characterization of the nature of the pure transition is needed in order to explore the physics of the dilute model we find clear indication of a first order transition at tt 0 62731 or 0 00006 using a cluster method that can be extended to the dilute case and histogram reweighting analysis our estimate is two orders of magnitude more precise than previous simulations and falls within the error bounds of a reanalysis of existing series expansion tt 0 628 or 0 004 reported on herein

an infinity of phase transitions as a function of temperature exact results for a model with fixed point imaging abstract position space renormalization group methods are used to derive exact results for an ising model on a fractal lattice the model incorporates both nearest neighbor and long range interactions the long range interactions which span all length scales on the lattice can be thought of as resulting from fractal periodic boundary conditions we present exact phase diagrams and specific heats in terms of these two interactions and show that a hall of mirrors fixed point imaging mechanism leads to an infinite number of phase transitions

decomposition of higher order wright convex functions in this paper we consider higher order wright convex functions and prove that they are representable as the sum of a continuous higher order convex function and a polynomial function

retarded hydrodynamic interactions in suspensions abstract we extend the method to determine the grand mobility matrix presented in a previous article to the problem of two particles in an unbounded fluid satisfying the time dependent linear incompressible navier stokes equations use of this method combined with a generalization of the fluctuation dissipation theorem results in the determination of the correlation matrix we determine the effect of hydrodynamic interactions on the time dependent diffusion coefficients d 11 and d 12 finally we study the effects of the correlation functions on brownian motion

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