hierarchical pinning models quadratic maps and quenched disorder we consider a hierarchical model of polymer pinning in presence of quenched disorder introduced by derrida et al j stat phys 66 1189 1213 1992 which can be re interpreted as an infinite dimensional dynamical system with random initial condition the disorder it is defined through a recurrence relation for the law of a random variable r n n 1 2 which in absence of disorder i e when the initial condition is degenerate reduces to a particular case of the well known logistic map the large n limit of the sequence of random variables 2 n log r n a non random quantity which is naturally interpreted as a free energy plays a central role in our analysis the model depends on a parameter 0 1 related to the geometry of the hierarchical lattice and has a phase transition in the sense that the free energy is positive if the expectation of r 0 is larger than a certain threshold value and it is zero otherwise it was conjectured in derrida et al j stat phys 66 1189 1213 1992 that disorder is relevant respectively irrelevant or marginally relevant if 1 2 1 2 we find the correct scaling form for weak disorder of the critical point shift

ising models julia sets and similarity of the maximal entropy measures we study the phase transition of ising models on diamondlike hierarchical lattices following an idea of lee and yang one can make an analytic continuation of free energy of this model to the complex temperature plane it is known that the migdal kadanoff renormalization group of this model is a rational endomorphism denoted byf of and that the singularities of the free energy lie on the julia setj f the aim of this paper is to prove that the free energy can be represented as the logarithmic potential of the maximal entropy measure onj f moreover using this representation we can show a close relationship between the critical exponent and local similarity of this measure

on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

electrical circuits on mesoscopic sierpinski gaskets a deterministic network of impedances on the sierpinski gasket fractal is considered with particular regard to low generation or mesoscopic systems we study the fractal set of the resonances of the circuit and present calculations on frequency dependent systems the condition of an intermediate size can lead to an oscillatory scaling of the total impedance that disappears in the asymptotic limit several examples are given

mobius mellin and mathematical physics abstract we examine some results and techniques of analytic number theory which have application or potential application in mathematical physics we consider inversion formulae for lattice sums various transformations of infinite series and products functional equations and scaling relations with selected applications in electrostatics and statistical mechanics in the analysis the mellin transform and the riemann zeta function play a key role

the asymmetric clock model on a cayley tree the asymmetric clock model is studied on a cayley tree and commensurate and incommensurate phases are found the transition lines are obtained from stability conditions and characteristic points in the phase diagram are analysed by numerical iteration a critical endpoint is found for the case of three states per site and a lifshitz point for the case of four states per site

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

chaotic properties of multipoint correlation functions of an ising model with long range interactions on the sierpinski gasket lattice a class of multispin correlation functions of an ising model with ferromagnetic nearest neighbor interactionsk and constant distance independent long range interactionsq1 q l 1 2 on the sierpinski gasket lattice is considered using an exact method for calculating thermodynamic functions of hierarchically constructed ising systems it is shown that for a set of values ofq and for almost all values ofk somemk spin correlation functions wheremk 3 k 3 withk 1 2 n andn 1 2 being the order of lattice construction change chaotically asn k and therebymk increase to infinity accordingly in the thermodynamic limit these correlation functions prove to be nonanalytic for appropriate values ofq andk sincemk point correlation functions withk being finite i e correlation functions involving finite numbers of spins remain analytic asn tends to infinity there is a smooth crossover between analytic properties of correlation functions of the two types

a rescaling approach to two and three dimensional lattice spectra an exact real space rescaling transformation is used to calculate the spectral properties of tight binding hamiltonians with an arbitrary range of interaction of two and three dimensional lattices the intrinsic problem of the proliferation of initial short range interactions under rescaling is shown to be tractable numerically this method provides a direct way of obtaining the local green functions and local density of states singularities of the spectral functions are associated with fixed points or limit cycles of the rescaling transformation band edge fixed points and exponents have been found for the square triangular and cubic lattices in two and three dimensions there is rapid convergence to accurate numerical results for the green function outside the band while good results have been obtained for the in band e g density of states spectra in two dimensions by suitable choices of cluster size and attenuation factor insight into the relationship between these factors follows from an investigation of finite cluster approximations to the exact transformation it is found that the zeros of the green functions in the complex plane are julia sets which have a fractal structure for a given cluster size the attenuation factors must lie outside these zeros in order for the procedure to be convergent

hierarchical ensembles random fractals flow fractals and the renormalisation group hierarchical ensembles possess exactly renormalisable partition functions examples studied in this work are iterative geometric constructions which at criticality give random self similar fractals the ensembles include novel non self similar constructions termed flow fractals here the simple criticality of these ensembles is analysed and the self similar fractals are found to be homogeneous a multiple weight ensemble the ising snowflake is introduced the transfer matrix of fractals is defined via the renormalisation group a discussion and interpretation is given of the eigenvalues of the transfer matrix fractals for the ising snowflake

infinite susceptibility at high temperatures in the migdal kadanoff scheme the migdal kadanoff real space renormalisation group scheme applied to the ising and classical xy and probably other ferromagnetic models yields an infinite susceptibility at all temperatures above the critical temperature for a hypercubic lattice of dimension d or 2

tutte polynomial of scale free networks the tutte polynomial of a graph or equivalently the q state potts model partition function is a two variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics the computation of this invariant for a graph is np hard in general in this paper we focus on two iteratively growing scale free networks which are ubiquitous in real life systems based on their self similar structures we mainly obtain recursive formulas for the tutte polynomials of two scale free networks lattices one is fractal and large world while the other is non fractal but possess the small world property furthermore we give some exact analytical expressions of the tutte polynomial for several special points at x y plane such as the number of spanning trees the number of acyclic orientations etc

ac properties of 3d sierpinski gaskets rigorous results the problem of the decimation of a network of impedances on the three dimensional sierpinski gasket is solved the exact map is given and its asymptotic behaviours are studied the most significant invariant subspaces of and the associated submaps are considered this also allows us to address the problem of small size phenomena such as oscillating asymptotic behaviour on this kind of fractal the set of the resonances of the system and the frequency dependence of the total impedance are studied both in the thermodynamic limit and in mesoscopic systems

asymptotics of the susceptibility for the ising model on the hierarchical lattices using the gks inequality griffiths and kaufman showed in 12 that the susceptibility in the ising model on the hierarchical lattices is infinite fort tc in the present work we give an analytic calculation of the dominant singularity of the susceptibility whenh 0 fort tc moreover we analytically find the spontaneous magnetization fort tc and prove that fort tc the susceptibility is finite

renormalisation groups with periodic and aperiodic orbits motivated by recent results for some frustrated hierarchical models the authors analyse the consequences of renormalisation group transformations with periodic and aperiodic orbits they find that the free energy has in general an infinite number of singularities this occurs already when the transformation has a stable periodic orbit they present an idealised example and a new class of hierarchical potts models to which the theory applies

ising model in an external field on a hierarchical lattice the two dimensional renormalization map of the diamond hierarchical ising model in an external field is given and pictures of the distribution of zeros of the partition function in the complex plane of temperature for varying values of coupling constant and external field are shown critical exponents of the model are found and results are different from those of the ising model on a two or three dimensional regular lattice

one dimensional mapping modulated phases and lyapunov exponent for the antiferromagnetic q state potts and multi site exchange interaction ising models abstract we describe the bifurcation structure period doubling and chaos for the antiferromagnetic q state potts model on the bethe lattice and three site interaction ising model on husimi one in a magnetic field by using the recursion relation technique a chaotic behavior of the magnetic susceptibility for the models is observed at low temperatures the resulting one dimensional rational mapping has a positive lyapunov exponent in the region of the chaotic regime for the antiferromagnetic q state potts q 2 and three site interaction ising models we discuss modulated phases for the antiferromagnetic q state potts q 2 and q 2 and three site interaction ising model at low temperatures the q state potts model q 2 has only one modulated phase with 1 2 pinching corresponding to the 2 cycle the q state potts q 2 and three site interaction ising models have an infinite number of modulated phases with different pinching numbers we construct the first modulated phase after the first bifurcation point

3d magnetotelluric inversion using a limited memory quasi newton optimization the limited memory quasi newton method with simple bounds is used to develop a novel fully 3d magnetotelluric mt inversion technique this nonlinear inversion is based oniterativeminimizationofaclassicaltikhonovregularized penalty function however instead of the usual model space of log resistivities the approach iterates in a model space with simple bounds imposed on the conductivities of the 3d target the method requires storage proportional to 2 ncp n wherenisthenumberofconductivitiestoberecovered and ncp is the number of correction pairs practically only a few these requirements are much less than those imposed byothernewtonmethods whichusuallyrequirestorageproportional ton m orn n where m is the number of data tobeinverted thederivativesofthepenaltyfunctionarecalculated using an adjoint method based on electromagnetic fieldreciprocity theinversioninvolvesallfourentriesofthe mt impedance matrix the x3d integral equation forwardmodeling code is used as an engine for this inversion convergence performance and accuracy of the inversion are demonstrated on synthetic numerical examples after investigating erratic resistivities in the upper part of the model obtainedforoneoftheexamples weconcludethatthestandard tikhonov regularization is not enough to provide consistently smooth underground structures an additional regularizationhelpstoovercometheproblem

asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors the present paper concerns the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a one dimensional heat conductive hydrodynamic model for semiconductors it is important to analyze thermal influence on the motion of electrons in semiconductor device to improve the reliability of devices by handling a hot carrier problem we show the unique existence of the stationary solution satisfying a subsonic condition by using the leray schauder and the schauder fixed point theorems then the asymptotic stability of the stationary solution is proved by deriving the a priori estimate uniformly in time here an energy form plays an essential role we also prove that the solution converges to the stationary solution exponentially fast as time tends to infinity

ferromagnetic phase transitions of inhomogeneous systems modelled by square ising models with diamond type bond decorations abstract the two dimensional ising model defined on square lattices with diamond type bond decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems the model is studied analytically under the bond renormalization scheme for an n level decorated lattice the long range ordering occurs at the critical temperature given by the fitting function k b t c j n 1 6410 0 6281 exp 0 5857 n and the local ordering inside n level decorated bonds occurs at the temperature given by the fitting function k b t m j n 1 6410 0 8063 exp 0 7144 n the critical amplitude a sin g n of the logarithmic singularity in specific heat characterizes the width of the critical region and it varies with the decoration level n as a sin g n 0 2473 exp 0 3018 n obtained by fitting the numerical results the cross over from a finite decorated system to an infinite decorated system is not a smooth continuation for the case of infinite decorations the critical specific heat becomes a cusp with the height c n 0 639852 the results are compared with those obtained in the cell decorated ising model

3d inversion for multipulse airborne transient electromagnetic data abstractmultipulse airborne transient electromagnetic atem systems transmit one high power pulse and one low power pulse containing more high frequency em signals such systems have better near surface resolutions while maintaining the depth of exploration of other conventional systems atem systems are especially suitable for geologic mapping and mineral exploration the inversion of multipulse atem data has been mainly limited to 1d modeling which is not suitable for complex underground structures we have investigated an algorithm for 3d multipulse atem data inversion based on direct gauss newton optimization with quite fast convergence the forward problems were solved in the frequency domain based on the secondary scattered electrical field equation and then the inverse fourier transform and the convolution with transmitting waveform were applied to calculate the arbitrary waveform response and sensitivity matrix in the time domain to optimize the number of computations and memory we further us

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

meblin transforms associated with julia sets and physical applications we introduce the mellin transform of the balanced invariant measure associated to the julia set generated by a rational transformation we show that its analytic continuation is a meromorphic function the poles of which are on a semi infinite periodic lattice this allows one to have an understanding of the behavior of the measure near a repulsive fixed point trace identities corresponding to the fact that the analytically continued mellin transform vanishes at negative integers are derived for the polynomial case the quadratic map is first analyzed in detail and the analytic properties of the inverse of the green s function are exhibited of interest is the appearance of a dense set of spikes at dyadic points when the julia set is disconnected these results are used to study the residues of the mellin transform a certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems the excitation spectrum of which displays unusual oscillations the appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion

diffusion in a one dimensional lattice with random asymmetric transition rates the authors study one dimensional continuous time random walks for which the pairs wn wn 1 of nearest neighbour transition rates are assumed to be independent equally distributed random variables the long time asymptotic behaviour of the mean displacement x t is determined exactly for a specific model system in which diodes u 0 and two way bonds lambda v v occur with probabilities p and 1 p respectively for lambda 1 p we find that x t approximately tnu f beta 1ln t where nu ln 1 p ln lambda and beta ln lambda and where f is a periodic function with period 1 the mean displacement thus not only increases slower than linearly in time but exhibits superimposed non decaying oscillations

bifurcation analysis of the degond lucquin desreux morrow model for gas discharge abstract the main purpose of this paper is to investigate mathematically gas discharge townsend discovered and mechanisms which are essential for ionization of gas and then derived a threshold of voltage at which gas discharge can happen in this derivation he used some simplification such as discretization of time therefore it is an interesting problem to analyze the threshold by using the degond lucquin desreux morrow model and also to compare the results of analysis with townsend s theory note that gas discharge never happens in townsend s theory if mechanism is not taken into account in this paper we study an initial boundary value problem to the model with mechanism but no mechanism this problem has a trivial stationary solution of which the electron and ion densities are zero it is shown that there exists a threshold of voltage at which the trivial solution becomes unstable from stable then we conclude that gas discharge can happen for a voltage greater than this threshold even if mechanism is not taken into account it is also of interest to know the asymptotic behavior of solutions to this initial boundary value problem for the case that the trivial solution is unstable to this end we establish bifurcation of non trivial stationary solutions by applying crandall and rabinowitz s theorem and show the linear stability and instability of those non trivial solutions

distribution of repetitions of ancestors in genealogical trees we calculate the probability distribution of repetitions of ancestors in a genealogical tree for simple neutral models of a closed population with sexual reproduction and non overlapping generations each ancestor at generation g in the past has a weight w which is up to a normalization the number of times this ancestor appears in the genealogical tree of an individual at present the distribution pg w of these weights reaches a stationary shape p1 w for large g i e for a large number of generations back in the past for small w p1 w is a power law p1 w w with a non trivial exponent which can be computed exactly using a standard procedure of the renormalization group approach some extensions of the model are discussed and the eect of these variants on the shape of p1 w are analysed c 2000 elsevier science b v all rights reserved

remarks on disorder and aperiodicity in a model for interacting polymers abstract we present a comparative study of the effects of random and aperiodically distributed interactions on the critical behavior of a model for two interacting polymers on a diamond hierarchical lattice the problem is formulated in terms of exact renormalization group rg recursion relations in the disordered case it is possible to develop a perturbative treatment in order to obtain the fixed points of the moments associated with the random distribution of interactions fully uncorrelated disorder may become relevant driving the system away from a homogeneous fixed point layered disorder may lead to a breakdown of the perturbative treatment in the case of aperiodic interactions we also show some examples of relevance and irrelevance of geometric fluctuations and further investigate the models by resorting to an independent transfer matrix tm analysis which fully corroborates the scaling results

log periodic critical amplitudes a perturbative approach log periodic amplitudes appear in the critical behavior of a large class of systems in particular when a discrete scale invariance is present here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial in this case the log periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated

julia sets and complex singularities in hierarchical ising models we study the analytical continuation in the complex plane of free energy of the ising model on diamond like hierarchical lattices it is known 12 13 that the singularities of free energy of this model lie on the julia set of some rational endomorphismf related to the action of the migdal kadanoff renorm group we study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin off we prove that for almost all with respect to the harmonic measure geodesics the complex critical exponent is common and compute it

nonlinear conjugate gradients algorithm for 2 d magnetotelluric inversion we investigate a new algorithm for computing regularized solutions of the 2 d magnetotelluric inverse problem the algorithm employs a nonlinear conjugate gradients nlcg scheme to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity we compare this algorithm theoretically and numerically to two previous algorithms for constructing such minimum structure models the gauss newton method which solves a sequence of linearized inverse problems and has been the standard approach to nonlinear inversion in geophysics and an algorithm due to mackie and madden which solves a sequence of linearized inverse problems incompletely using a linear conjugate gradients technique numerical experiments involving synthetic and field data indicate that the two algorithms based on conjugate gradients nlcg and mackie madden are more efficient than the gauss newton algorithm in terms of both computer memory requirements and cpu time needed to find accurate solutio

fine structure and complex exponents in power law distributions from random maps discrete scale invariance dsi has recently been documented in time to failure rupture earthquake processes and financial crashes in the fractal geometry of growth processes and in random systems the main signature of dsi is the presence of log periodic oscillations correcting the usual power laws corresponding to complex exponents log periodic structures are important because they reveal the presence of preferred scaling ratios of the underlying physical processes here we present new evidence of log periodicity overlaying the leading power law behavior of probability density distributions of affine random maps with parametric noise the log periodicity is due to intermittent amplifying multiplicative events we quantify precisely the progressive smoothing of the log periodic structures as the randomness increases and find a large robustness our results provide useful markers for the search of log periodicity in numerical and experimental data

data adaptive wavelets and multi scale singular spectrum analysis using multi scale ideas from wavelet analysis we extend singular spectrum analysis ssa to the study of nonstationary time series including the case where intermittency gives rise to the divergence of their variance the wavelet transform resembles a local fourier transform within a finite moving window whose width w proportional to the major period of interest is varied to explore a broad range of such periods ssa on the other hand relies on the construction of the lag correlation matrix c onm lagged copies of the time series over a fixed window width w to detect the regular part of the variability in that window in terms of the minimal number of oscillatory components herew dm1t with1t as the time step the proposed multi scale ssa is a local ssa analysis within a moving window of widthm w n wheren is the length of the time series multi scale ssa variesw while keeping a fixed w m ratio and uses the eigenvectors of the corresponding lag correlation matrix c m as data adaptive wavelets successive eigenvectors of c m correspond approximately to successive derivatives of the first mother wavelet in standard wavelet analysis multi scale ssa thus solves objectively the delicate problem of optimizing the analyzing wavelet in the time frequency domain by a suitable localization of the signal s correlation matrix we present several examples of application to synthetic signals with fractal or power law behavior which mimic selected features of certain climatic or geophysical time series the method is applied next to the monthly values of the southern oscillation index soi for 1933 1996 the soi time series is widely believed to capture major features of the el nino southern oscillation enso in the tropical pacific our methodology highlights an abrupt periodicity shift in the soi near 1960 this abrupt shift between 5 and 3 years supports the devil s staircase scenario for the enso phenomenon preliminary results of this study were presented at the xxii general assembly of the european geophysical society vienna may 1997 and at the fall meeting of the american geophysical union san francisco december 1997 2000 elsevier science b v all rights reserved

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

continuity of julia set and its hausdorff dimension of yang lee zeros abstract we show the continuity of the julia set and its hausdorff dimension about a family of rational maps concerning 2 dimensional diamond hierarchical potts models about anti ferromagnetic coupling in statistical mechanics

hierarchical lattice with competing interactions an example of a nonlinear map an ising model with competing nearest and more than nearest neighbour interactions on a hierarchical lattice is solved by decimation in the context of nonlinear mappings the renormalisation group trajectories are analysed as the competition parameter is varied commensurate and incommensurate phases are identified

dynamical properties of quasi crystals fibonacci chain and penrose lattice the discrete scale invariance of quasi periodic systems leads to a scaling relationship omega l zf l between a characteristic dynamic variable omega and length l where f l is a periodic function of ln l the dynamic exponent z can be calculated for diffusion spin wave and phonon dynamics where omega is respectively gamma omega omega 2 where gamma and omega are characteristic rates or frequencies respectively by exploiting a crossover argument which results in z df t where df is the fractal dimension and t is the length scaling exponent for the conductance the latter exponents and hence z are calculated for a fibonacci chain model df t z 1 1 2 and via an exact bond moving technique for the penrose lattice df t z 2 0 2 the technique also provides the thermal exponent of the heisenberg spin model on the two quasi crystals nu 1 infinity respectively

on connectivity of julia sets of yang lee zeros we prove that the julia sets of the yang lee zeros of the potts model on the diamond hierarchical lattice are connected sets on the complex plane

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

critical properties of an aperiodic model for interacting polymers we investigate the effects of aperiodic interactions on the critical behaviour of an interacting two polymer model on hierarchical lattices equivalent to the migadal kadanoff approximation for the model on bravais lattices via renormalization group and transfer matrix calculations the exact renormalization group recursion relations always present a symmetric fixed point associated with the critical behaviour of the underlying uniform model if the aperiodic interactions defined by substitution rules lead to relevant geometric fluctuations this fixed point becomes fully unstable giving rise to novel attractors of different nature we present an explicit example in which this new attractor is a two cycle attractor with critical indices different from the uniform model in the case of the four letter rudin shapiro substitution rule we find a surprising closed curve whose points are attractors of period two associated with a marginal operator nevertheless a scaling analysis indicates that this attractor may lead to a new critical universality class in order to provide an independent confirmation of the scaling results we turn to a direct thermodynamic calculation of the specific heat exponent the thermodynamic free energy is obtained from a transfer matrix formalism which had been previously introduced for spin systems and is now extended to the two polymer model with aperiodic interactions

mapping between hierarchical lattices by renormalisation and duality the renormalisation group scheme of migdal 1976 and kadanoff 1976 is applied to a ferromagnetic potts model on two hierarchical lattices in the two cases the renormalisation transformation is split into two elementary processes parallel bonds or series bonds and each lattice can be mapped onto the other one it is then easy to deduce the characteristic parameters critical exponents repulsive fixed points etc of one lattice from those of the other lattice moreover the classical geometrical duality offers a third possibility of mapping and the relationship between the renormalisation and the duality methods is presented in the peculiar case of a two dimensional lattice

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

electrical properties of percolation clusters exact results on a deterministic fractal a recursively built deterministic fractal lattice generalising a model proposed by kirkpatrick 1979 is used to examine general electrical properties of percolation clusters in arbitrary dimensions most physical quantities are exactly analytically tractable since the model admits an exact renormalisation group transformation associated with a rational mapping t x of one complex variable the model has one free parameter f the fraction of conducting material which is put equal to the self dual point f 2 3 in two dimensions and chosen to reproduce the numerical values of the exponent ratio t nu in higher dimensions results concern in particular the frequency dependence of the impedance and the loss angle where the authors prove the existence of scaling laws at low and high frequency they also determine exactly the transient response to an arbitrary input signal and relate the distribution of relaxation times of the infinite lattice to well known mathematical objects associated to the rational transform t namely its julia set and its invariant measure the critical amplification of flicker or 1 f resistor noise is also considered it is shown to obey scaling laws with its own critical exponents in agreement with other recent works the relationships to other theoretical models and to experiments on metal insulator mixtures are discussed

fisher zeros and julia sets a multifractal analysis abstract we consider the potts model on a hierarchical lattice a multifractal analysis of the julia set associated with the renormalization group transformation which gives the fisher zeros of the partition function in the complex temperature plane is performed the behavior of the singularity spectrum f and the generalized dimension dq as a function of the number of states is studied

strange objects in the complex plane julia sets are examined as examples of strange objects which arise in the study of long time properties of simple dynamical systems technically they are the closure of the set of unstable cycles of analytic maps physically they are sets of points which lead to chaotic behavior the mapf z z2 p is analyzed for smallp where the julia set is a closed curve and for largep where the julia set is completely disconnected in both cases the hausdorff dimension is calculated in perturbation theory and numerically an expression for the rate at which points escape from the neighborhood of the julia set is derived and tested in a numerical simulation of the escape

a real space renormalisation group approach to electrical and noise properties of percolation clusters the migdal kadanoff renormalisation scheme which becomes exact on hierarchical lattices is used to investigate electrical properties of 2d and 3d percolation clusters the author studies in detail the frequency dependence of the conductivity and the loss angle and the amplification of resistance flicker noise this type of noise exhibits critical amplification with its own exponents x and y which are related to the exponent b recently defined by rammal et al 1985

renormalization group analysis of the 2000 2002 anti bubble in the us s p500 index explanation of the hierarchy of five crashes and prediction abstract we propose a straightforward extension of our previously proposed log periodic power law model of the anti bubble regime of the usa stock market since the summer of 2000 in terms of the renormalization group framework to model critical points using a previous work by gluzman and sornette phys rev e 65 2003 036142 on the classification of the class of weierstrass like functions we show that the five crashes that occurred since august 2000 can be accurately modeled by this approach in a fully consistent way with no additional parameters our theory suggests an overall consistent organization of the investors forming a collective network which interact to form the pessimistic bearish anti bubble regime with intermittent acceleration of the positive feedbacks of pessimistic sentiment leading to these crashes we develop retrospective predictions that confirm the existence of significant arbitrage opportunities for a trader using our model finally we offer a prediction for the unknown future of the us s p500 index extending over 2003 and 2004 that refines the previous prediction of sornette and zhou quant finance 2 2002 468

a new real space renormalisation method and its julia set the authors introduce a new real space renormalisation method which is valid not only on the real axis but also everywhere in the complex plane of temperatures the method requires the knowledge of the partition function on small lattices and can in principle be improved systematically they test this method on the two dimensional ising model they find that it gives a satisfactory picture for the set of zeros of the partition function when restricted to the real axis the estimates for the renormalisation transformation the critical point and the thermal exponent are also very good

restoration of macroscopic isotropy on d 1 simplex fractal conductor networks abstract restoration of macroscopic isotropy has been investigated in d 1 simplex fractal conductor networks via exact real space renormalization group transformations using some theorems of fixed point theory it has been shown very rigoroursly that the macroscopic conductivity becomes isotropic for large scales and anisotropy vanishes with a scaling exponent which is computed exactly for arbitrary values of d and decimation numbers b 2 3 4 and 5

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

oscillatory critical amplitudes in hierarchical models and the harris function of branching processes oscillatory critical amplitudes have been repeatedly observed in hierarchical models and in the cases that have been taken into consideration these oscillations are so small to be hardly detectable hierarchical models are tightly related to iteration of maps and in fact very similar phenomena have been repeatedly reported in many fields of mathematics like combinatorial evaluations and discrete branching processes it is precisely in the context of branching processes with bounded off spring that t harris in 1948 first set forth the possibility that the logarithm of the moment generating function of the rescaled population size in the super critical regime does not grow near infinity as a power but it has an oscillatory prefactor the harris function these oscillations have been observed numerically only much later and while the origin is clearly tied to the discrete character of the iteration the amplitude size is not so well understood the purpose of this note is to reconsider the issue for hierarchical models and in what is arguably the most elementary setting the pinning model that actually just boils down to iteration of polynomial maps and notably quadratic maps in this note we show that the oscillatory critical amplitude for pinning models and the harris function coincide moreover we make explicit the link between these oscillatory functions and the geometry of the julia set of the map making thus rigorous and quantitative some ideas set forth in derrida et al commun math phys 94 115 132 1984

anomalous diffusion on fractal lattices with site disorder like random walks on euclidean lattices random walks on fractal lattices are modified by a waiting time site disorder when the first moment of the waiting time distribution diverges it is shown that for lattices which support recurrent walks spectral dimension smaller than two the inverse of the diffusion exponent in the presence of disorder is increased by the difference between the waiting time fractal dimension and the usual fractal dimension this hyperscaling relation is derived for sierpinski gaskets in arbitrary dimension with scale dependent waiting times this provides qualitative insight into this problem and the exponent relation derived should also hold for statistically self similar structure such as percolation clusters for lattices whose usual spectral dimension is larger than two a mean field result holds

renormalisation group study of a disordered model we study the potts model on a diamond hierarchical lattice with random interactions using weak disorder expansions we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive at ap 0 we find how the logarithmic singularity is modified by the disorder lastly we suggest that this model should present griffiths like singularities

singular behavior of the density of states and the lyapunov coefficient in binary random harmonic chains we study the integrated density of statesh 2 of a chain of harmonic oscillators with a binary random distribution of the masses we show in particular that there is a dense set of values of the squared frequency for which the differenceh 2 h 2 has a singularity of the type 2 multiplied by a periodic function of ln where the exponent and the period depend continuously on 2 in the region where 1 2 h is not differentiate on a dense set of points the same type of singularities is also present in the lyapunov coefficient