probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

supersymmetric quantum mechanics with levy disorder in one dimension we consider the schrodinger equation with a random potential of the form open image in new window where w is a levy noise we focus on the problem of computing the so called complex lyapunov exponent open image in new window where n is the integrated density of states of the system and is the lyapunov exponent in the case where the levy process is non decreasing we show that the calculation of reduces to a stieltjes moment problem we ascertain the low energy behaviour of the density of states in some generality and relate it to the distributional properties of the levy process we review the known solvable cases where can be expressed in terms of special functions and discover a new one

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

twenty five years of multifractals in fully developed turbulence a tribute to giovanni paladin the paper on the multifractal nature of fully developed turbulence and chaotic systems by benzi et al 1984 j phys a math gen 17 3521 has been a starting point of many investigations on the different faces of self similarity and intermittency in turbulent phenomena since then the multifractal model has become a useful tool for the study of small scale turbulence in particular for detailed predictions of different eulerian and lagrangian statistical properties on the occasion of the 50th birthday of our unforgettable friend and colleague giovanni paladin 1958 1996 we review here the basic concepts and some applications of the multifractal model for turbulence

cascade structures and scaling exponents in a dynamical model of turbulence measurements and comparison a detailed examination of the cascade statistics and scaling exponents is carried out for a dynamical system model of fully developed turbulence called the goy shell model the convergence in time of the probability density functions and moments of the velocity fluctuations and their scaling exponents is studied with particular care with a large sample size 5 ifmmode times else texttimes fi 10 9 we demonstrate that there exists a finite cutoff for the velocity fluctuations at each inertial range wave number shell and the properties of the cutoff determine the scaling exponents of all moments this cutoff represents the most intermittent structures in the cascade dynamics and exhibits a power law dependence on wave number the accurately determined scaling exponents permit a detailed comparison with various phenomenological models describing the statistics of the energy cascade the consideration of the first and second derivatives of the scaling exponents with respect to the order of the moments p provides the evidence that the hierarchical structure model she and leveque phys rev lett 72 336 1994 predicts the best functional dependence on p of the scaling exponents in the goy shell model

products of random matrices and generalised quantum point scatterers to every product of 2 2 matrices there corresponds a one dimensional schrodinger equation whose potential consists of generalised point scatterers products of random matrices are obtained by making these interactions and their positions random we exhibit a simple one dimensional quantum model corresponding to the most general product of matrices in sl 2 we use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms

predictability of velocity and temperature fields in intermittent turbulence we discuss the relation between predictability and the sensitive dependence on initial conditions in turbulent flows the dependence on the reynolds number of the maximum lyapunov exponent of the flow is evaluated within the multifractal model the temporal intermittency of the degree of chaos is found to localize on the degrees of freedom corresponding to high wavenumbers we discuss the consequences for the mechanism for the growth of small perturbations we show that intermittency causes long tails in the distribution of the predictability times both in the velocity field and in a passively advected scalar field though there is no simple relation between these two times numerical evidence of this picture is provided within the framework of a cascade model for velocity and temperature fields in fully developed turbulence

on intermittency in a cascade model for turbulence in this note we study the possibility of performing analytic computations of the exponents characterizing the multifractal behaviour of turbulence a simple analytic computation is presented in the framework of the cascade model or shell model

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

parameter estimation in non extensive thermostatistics abstract equilibrium statistical physics is considered from the point of view of statistical estimation theory this involves the notions of statistical model of estimators and of exponential family a useful property of the latter is the existence of identities obtained by taking derivatives of the logarithm of the partition sum it is shown that these identities still exist for models belonging to generalised exponential families in which case they involve escort probability distributions the percolation model serves as an example a previously known identity is derived it relates the average number of sites belonging to the finite cluster at the origin the average number of perimeter sites and the derivative of the order parameter

universality and superuniversality of multifractals in nonlinear resistor networks the multifractal functionf is generalized to describe noisy nonlinear random resistor networks an approximant function for the family of noise exponents is introduced that provides a good description of real percolative systems for strong nonlinearities by mapping from this family to the multifractal function one can approximate the latter a scale transformation of in the approximation makes the multifractal function universal for all nonlinearities and by applying an additional transformation this function becomes superuniversal i e independent of the dimension the universality is demonstrated for the mandelbrot given structure and the implications of these results are discussed on real percolative systems

series expansion study of the distribution of currents in the elements of a random diode insulator network the critical exponent zeta k of the kth moment of the current distribution for random diode insulator networks on the square and simple cubic lattices is calculated for a range of k using low density series expansion techniques it is also shown that zeta 1 nu sub where nu sub is the critical exponent of the parallel connectedness length for directed percolation the values of zeta k for the simple cubic lattice are well fitted by a simple exponential formula with zeta k 1 zeta k 1 1 1 2 the skal shklovskii scaling relation for the conductivity is generalised to the kth moment and it follows that the exponent k describing the divergence of flicker noise is given by kappa approximately d 1 nu perpendicular to nu sub zeta 4 2 zeta 2 the authors estimate this for both lattices

lyapunov exponents one dimensional anderson localization and products of random matrices the concept of lyapunov exponent has long occupied a central place in the theory of anderson localisation its interest in this particular context is that it provides a reasonable measure of the localisation length the lyapunov exponent also features prominently in the theory of products of random matrices pioneered by furstenberg after a brief historical survey we describe some recent work that exploits the close connections between these topics we review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case finally we point out some limitations of the lyapunov exponent as a means of studying localisation properties

multifractality and the kauffman model the author simulates the spread of an initial damage in the kauffman cellular automata at its critical point on the square lattice he calculates various moments of the probability that a site has been damaged n times and checks for multifractality in the critical exponents for different ensembles specifically he finds no evidence for multifractality when the moments of the probabilities are evaluated with lattice size l but multifractal behaviour occurs when the moments are monitored as a function of time t

on the f alpha spectrum of one hump maps based on a perturbative procedure developed to solve for the universal parameters of one hump maps the corresponding fractal dimensions dq and the f alpha spectrum are studied

self avoiding random walks on multifractal lattices a renormalisation theory is developed to study the critical behaviour of self avoiding random walks on multifractals critical exponents and connectivity constants are calculated for walks on a class of square multifractal lattices using a finite lattice renormali sation the effect of the multifractal disorder is considered for both annealed and quenched disorder

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

explicit invariant measures for products of random matrices we construct explicit invariant measures for a family of infinite products of random independent identically distributed elements of sl 2 c the matrices in the product are such that one entry is gamma distributed along a ray in the complex plane when the ray is the positive real axis the products are those associated with a continued fraction studied by letac seshadri z wahr verw geb 62 1983 485 489 who showed that the distribution of the continued fraction is a generalised inverse gaussian we extend this result by finding the distribution for an arbitrary ray in the complex right half plane and thus compute the corresponding lyapunov exponent explicitly when the ray lies on the imaginary axis the matrices in the infinite product coincide with the transfer matrices associated with a one dimensional discrete schrodinger operator with a random gamma distributed potential hence the explicit knowledge of the lyapunov exponent may be used to estimate the exponential rate of localisation of the eigenstates

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

harmonious representation of pdfs reflecting large deviations abstract the framework of multifractal analysis mfa is distilled to the most sophisticated one within this transparent framework it is shown that the harmonious representation of mfa utilizing two distinct tsallis distribution functions one for the tail part of probability density function pdf and the other for its center part explains the recently observed pdfs of turbulence in the highest accuracy superior to the analyses based on other models such as the log normal model and the p model

correlations in multifractals we present an analysis of the implicit assumptions that lie at the basis of the scaling relations for fractal measures proposed by cates et al 1 furthermore the scaling relations which have been shown to hold for trivial multifractals like the generalized cantor set 2 are tested numerically for critical wave functions of an incommensurate system

a new class of point interactions in one dimension we present a class of self adjoint extensions of the symmetric operator c 0 r1 0 which correspond formally to perturbations of the laplacian by pseudopotentials involving 2 these operators which provide new examples of generalized point interactions in the sense of seba are defined by the boundary conditions 0 e z 0 r 0 0 ez r 0 0 for z c r r we calculate their spectra resolvents and scattering matrices and show that they can be realized as limits of schrodinger operators with local short range potentials

intermittency as multifractality in history space temporal intermittency in a chaotic dynamical system can be regarded as a manifestation of the multifractal properties of the set of histories the renyi entropies kq which generalize the kolmogorov entropy k1 play a role analogous to the exponents tau q characterising the scaling properties of the moments of the mass distribution the authors introduce the topological entropy h lambda of the subset of histories with the same local expansion parameter they show that kq and h lambda are related to each other by means of a legendre transformation

multiscaling and multifractality abstract a general scaling argument is presented which predicts for the two points correlation function for growth phenomena and critical phenomena a scaling form multiscaling of the type g r r r d d r r a r r where r is the distance from the origin r a characteristic length and a x is an amplitude a connection between multifractality and multiscaling is shown the numerical data of plischke and racz on diffusion limited aggregation and a soluble model for spinodal decomposition are found in agreement with such multiscaling form

fluctuations of random matrix products and 1d dirac equation with random mass we study the fluctuations of certain random matrix products pi n m n cdots m 2m 1 of mathrm sl 2 mathbb r describing localisation properties of the one dimensional dirac equation with random mass in the continuum limit i e when matrices m n s are close to the identity matrix we obtain convenient integral representations for the variance gamma 2 lim n rightarrow infty mathrm var ln pi n n the case studied exhibits a saturation of the variance at low energy varepsilon along with a vanishing lyapunov exponent gamma 1 lim n rightarrow infty ln pi n n leading to the behaviour gamma 2 gamma 1 sim ln 1 varepsilon rightarrow infty as varepsilon rightarrow 0 our continuum description sheds new light on the kappus wegner band center anomaly

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

generalized kolmogorov entropy in the dynamics of multifractal generation abstract we point out that applying a maximization principle on a tsallis like generalized form of the kolmogorov entropy for iterated function systems naturally provides a canonical statistical frame for the description of the multifractal measures generated by such dynamical processes multifractal spectra can then be characterized by usual statistical parameters in particular the temperature we show that in the limit of zero temperature the multifractal measure collapses to a homogeneous distribution over a fractal support for finite temperatures multifractal spectra are studied numerically in an illustrative example

regular versus irregular laplacian growth multifractal spectroscopy the authors analyse the effect of tip stability on the multifractality of interfacial patterns by applying the method of noise reduction to a random laplacian growth on the square lattice while the distribution of perimeter sites over the harmonic measure is smooth in the case of random fractal objects with tip splitting separate peaks appear if the tips are stable these peaks can be identified with the subset of growth sites corresponding to the tips they demonstrate the consequences of such behaviour in the multifractal spectrum

multifractality of prime numbers abstract the multifractal formalism is applied to prime numbers the spectrum of critical indices is found to be contained in the interval min 1 where min tends to 1 for increasing sets of numbers besides the scaling of moments with respect to the length of intervals the scaling with respect to the sizes of subsets of natural numbers is also considered we have found the cusps in the plots of the functions and we claim that they are not caused by numerical roundings but they are a real effect besides the computer method some analytical calculations are presented

multiparticle diffusion limited aggregation with strip geometry abstract two finite density versions of the multiparticle dla model have been investigated on strips of width l lattice units with periodic boundary conditions in model i single particles are added to the deposit at the position in which they first attempt to step onto the deposit and in model ii particles directly and indirectly connected to the deposit via nearest neighbor occupancy are added when they first contact the growing deposit for model i the growth of the deposit mass crosses over from t 1 2 dependence in the low density short time limit to m t in the high density long time limit for model ii a similar crossover is found but as the density approaches the percolation threshold the deposit mass is found to grow according to m t with having a value of about 4 3 these results help to resolve some apparent discrepancies between the results of earlier work on closely related models by voss and by meakin and deutch

application of grobner bases to problems of movement of a particle abstract the classical method of grobner bases for multivariate polynomials in computer algebra and the related buchberger s algorithm and its modifications for the computation of such bases are applied to some elementary problems of kinematics as well as to the classical kepler newton problem in celestial mechanics where beyond the variables in the polynomials the differential operator d appears as well the popular computer algebra system maple v and the related standard package were used for this purpose and several possibilities of using grobner bases for the proof and or the derivation of formulae in mechanics are illustrated the present results generalize well known results for the proof derivation of geometric theorems by using classical grobner bases and related techniques and they illustrate the power of commercial computer algebra systems in the aforementioned tasks in kinematics modifications and generalizations of the present approach are also possible

novel dynamical scaling in kinetic growth phenomena abstract a novel dynamical scaling characterized by infinitely many growth exponents is presented this new behavior is manifested in the time dependent pair correlation function and therefore is amenable to direct experimental observation an exact analytical solution on the time dependent ginzburg landau model is obtained in the large n limit for a quench at zero temperature which exhibits such multiscaling behavior this new form of behavior reflects a morphological pattern in the growing domains and is expected in other growth phenomena as indicated by preliminary analysis on dla clusters a connection between multiscaling and multifractality is discussed

exact electronic spectra and inverse localization lengths in one dimensional random systems i random alloy liquid metal and liquid alloy analytic continuations into the complex energy plane of dyson schmidt type of equations for the calculation of the density of states are constructed for a random alloy model a liquid metal and for a liquid alloy in all these models the characteristic function follows from the solution of this equation its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions the equations have been solved exactly for some distributions of the random variables in the random alloy case the strengths of the delta potentials have an exponential distribution they may also have finite exponentially distributed values with probability 0 p 1 and be infinite with probability q 1 p in the liquid metal the liquid particles are assumed to behave like hard rods this implies an exponential distribution of the distances between the particles the common electronic potential may be arbitrary but is assumed to vanish outside the rods in the one dimensional liquid alloy there is apart from positional randomness of the liquid particles a distribution of the strengths of the electronic delta potentials for cauchy distributions an argument of lloyd is extended to obtain the characteristic function from the one in the model with equal strengths for the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above for several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed new results are found exponential decay of the density of states near special energies in the random alloy and liquid metal divergence of the density of states at certain energies with non classical exponent 13 in the random alloy if the average of the potential strengths vanishes exponentially small broadening of the bound state levels for low concentrations of the liquid particles peak in the localization length at the bound state energies which becomes exponentially narrow for low concentrations different exponent in the decay of the inverse localization length at large energies for delta potentials and square well potentials further an expression for the grand potential is given involving a sum over the characteristic function at certain points and divergence of the zero point energy is found for cauchy distributions of the delta potential strengths

nonlinear resistor fractal networks topological distances singly connected bonds and fluctuations the authors consider a fractal network of nonlinear resistors with the voltage v behaving as a power of the current i mod v mod r mod i mod alpha the resistance between two points at a distance l is r l varies as lzeta alpha they prove that zeta 0 describes the scaling of the topological chemical distance while zeta infinity describes that of the number of singly connected red bonds for random resistors they also consider the width of the resistance distribution delta r varies as l zeta 2 alpha values for zeta and zeta 2 are explicitly derived for two model fractals and delta r r is found to grow with l for the sierpinski gasket and alpha 1 612 the relevance of the results to percolation clusters is discussed

weak commutation relations and eigenvalue statistics for products of rectangular random matrices we study the joint probability density of the eigenvalues of a product of rectangular real complex or quaternion random matrices in a unified way the random matrices are distributed according to arbitrary probability densities whose only restriction is the invariance under left and right multiplication by orthogonal unitary or unitary symplectic matrices respectively we show that a product of rectangular matrices is statistically equivalent to a product of square matrices hereby we prove a weak commutation relation of the random matrices at finite matrix sizes which previously has been discussed for infinite matrix size moreover we derive the joint probability densities of the eigenvalues to illustrate our results we apply them to a product of random matrices drawn from ginibre ensembles and jacobi ensembles as well as a mixed version thereof for these weights we show that the product of complex random matrices yields a determinantal point process while the real and quaternion matrix ensembles correspond to pfaffian point processes our results are visualized by numerical simulations furthermore we present an application to a transport on a closed disordered chain coupled to a particle bath

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

computer based manipulation of systems of equations in elasticity problems with grobner bases abstract the classical method of buchberger s grobner bases for multivariate polynomial equations with a variety of applications in computer algebra is applied to three problems in elasticity concerning i a simple elastic truss ii an elementary plane elasticity problem in rectangular coordinates and iii the derivation of a compatibility partial differential equation for the stress components and the proof of biharmonicity of the same components also in plane elasticity the present results aim at the practical illustration of the possibility of deriving computer generated formulae in elasticity problems by using grobner bases as has been done in the case of geometric problems and the popular and modern computer algebra system maple v equipped with a related package has been used further generalizations of the present results are quite possible and some of these are discussed in brief

a new approach to the problem of disordered harmonic chains abstract in the problem of harmonic chains with random masses the exponential growth rate and the spectral distribution function are shown to be the real and imaginary part of dyson s characteristic function along its cut in the complex plane a new function satisfying a certain integral equation is introduced from which the characteristic function follows immediately the functions introduced by dyson schmidt and matsuda and ishii are recovered as special cases and it is shown that schmidt s distributions w n converge as n relations between dyson s and schmidt s functions are obtained finally the same method is applied to the problem of harmonic chains with random variables k n m n and k n 1 m n and to the tight binding electron model with random site energies

multifractal wavefunction at the localisation threshold the authors point out that the field theoretical description of the localisation transition exhibits anomalous scaling for the moments of the probability density at the localisation threshold this property hints at a multifractal structure of the wavefunction they attempt an analysis of the corresponding singularity distribution

universal power law tails for singularity dominated strong fluctuations integrals of the type i t identical to integral integral dxdy delta t h x y are considered where h is a function from an ensemble labelled by parameters i can represent circulation times of fluid particles in the plane orbital periods or semiclassical densities of states for one dimensional hamiltonian systems spectral densities for two dimensional crystals or the strength of a wave pulse produced by propagation of a deformed step discontinuity as h varies over the ensemble i develops strong fluctuations associated with legendre singularities for unrestricted functions h the probability distribution p i describing the fluctuations of i decays according to a universal i 9 law obtained by a scaling argument involving arnold s classification of catastrophes if h is only quadratic in y p varies as i 10 if the integral defining i is one dimensional p varies as i 3 if it is three dimensional i decays no faster than i 48 the probability distribution of i identical to mod delta i delta t mod decays as i 2

small scale turbulence in the plankton low order deterministic chaos or high order stochasticity abstract almost all regions of the oceans are heavily influenced by the effects of physical processes such as turbulence which in turn influence the distribution and ecology of organisms that occupy these regions there is a real need for additional focus placed on the precise knowledge of both physical and biological processes which is often difficult using basic time series analysis in that way we applied nonlinear analysis techniques to high frequency time series of temperature salinity and phytoplankton concentrations recorded in different hydrodynamical regimes related to tidal forcing in a tidally mixed coastal ecosystem techniques devoted to the identification of low order deterministic chaos cannot find evidence of chaos while a lower dimensionality was identified in low hydrodynamic conditions the results rather suggest stochastic time series with many degrees of freedom no obvious attractor in phase space trajectory lle indistinguishable from zero absence of convergence of the correlation integral we then applied to these data specific multifractal analysis techniques and showed that these time series clearly exhibit high order stochasticity in addition the stochastic structure of purely passive scalars i e temperature and salinity remained invariant while the one of phytoplankton biomass must be regarded as highly structured in time by both hydrodynamic and advective processes

multiscaling transformation in dynamical systems and turbulence abstract multiscaling is a scaling law where the exponent is slowing varying with the length scale pseudo algebraic law we discuss its origin as a consequences of multifractility and the existence of a lower cutoff in the calculation of correlation functions in different contexts we derive some exact results in the case of two scale cantor sets which can be extended to other fractal structures such as strange attractors of chaotic systems in fully developed turbulence the cutoff is naturally introduced by the viscosity and our approach leads to the prediction of an intermediate dissipation range which can be tested experimentally

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

maximal lyapunov exponents for random matrix products in this article we study the lyapunov exponent for random matrix products of positive matrices and express them in terms of associated complex functions this leads to new explicit formulae for the lyapunov exponents and to an efficient method for their computation

a connection between linear and nonlinear resistor networks the authors explore the connection between the higher moments of the current or voltage distribution in a random linear resistor network and the resistance of a nonlinear random resistor network they find that the two problems are very similar and that an infinite set of exponents are required to fully characterise each problem these exponent sets are shown to be identical on a particular hierarchical lattice a simple model which accurately describes the geometrical properties of the backbone of the infinite cluster at the percolation threshold and also the voltage distribution on this structure

multifractal moving average analysis and test of multifractal model with tuned correlations we address two common major problems in the study of time series characterizing fluctuations in complex systems multifractal analysis and multifractal modeling specifically we introduce a multi fractal centered moving average mf cma analysis which is computationally easier but equivalently performing compared with the well established multi fractal detrended fluctuation analysis mf dfa with linear detrending in addition we study in detail a generalized binomial multi fractal model gb mfm to conveniently and reliably generate multifractal surrogate data with arbitrary singularity strengths and arbitrary long term persistence we use the data generated by this model as well as realistic by construction monofractal data series with crossovers and trends to test and compare the multifractal analysis methods and discuss finite size effects as well as limitations due to spurious multifractality

are the cusps in the plots of f alpha a real effect an explicit example where the cusps in the plots of the f alpha spectrum appear is given an analysis is provided which gives the support for the claim that these cusps are not a computer artefact but are a signal of the breakdown of the scaling law

verification of the scaling relation within mpdft by analyzing pdfs for energy dissipation rates out of 40963 dns abstract the pdfs for energy dissipation rates extracted by kaneda and ishihara from their 40963 dns for fully developed turbulence are analyzed within the framework of multifractal probability density function theory mpdft to a high accuracy mpdft is a statistical mechanical ensemble theory for analyzing intermittent phenomena providing fat tail probability density functions pdfs recently mpdft has been improved by introducing a new scaling relation so as to be able to analyze intermittency through any series of pdfs with arbitrary magnification 1 it was fixed to a unique value 2 since the value of can be determined freely by observers the choice of should not affect observables the validity of the generalized mpdft is verified successfully through the precise analyses of several series of pdfs with different values of in the course of the verification it is revealed that the system of fully developed turbulence has much wider area representing scaling behaviors than the inertial range with the help of mpdft it has become possible to separate the coherent turbulent motion from fluctuations this may benefit the wavelet analysis of turbulence

renormalisation group for dla and fixed point distribution a renormalization group method is presented to analyse the multifractal structure of the growth probability in the diffusion limited aggregation dla a renormalisation group transformation is derived for the probability distribution of the growth bond s conductance after repeated scaling an invariant distribution is reached as a fixed point distribution the growth probability assigned to each growth bond is represented by a random multiplicative process of the cell s growth probability with fixed point distribution a hierarchy of generalised dimensions d q is calculated and the alpha f spectrum is found

a refutational approach to geometry theorem proving abstract a refutational method for proving universally quantified formulae in algebraic geometry is proposed a geometry statement to be proved is usually stated as a finite set of hypotheses implying a conclusion a hypothesis is either a polynomial equation expressing a geometric relation or a polynomial inequation the negation of a polynomial equation expressing a subsidiary condition that rules out degenerate cases and perhaps some general cases a conclusion is a polynomial equation expressing a geometry relation to be derived instead of showing that the conclusion directly follows from the hypothesis equations and inequations the proof by contradiction technique is employed it is checked whether the negation of the conclusion is inconsistent with the hypotheses this can be done by converting the hypotheses and the negation of the conclusion into a finite set of polynomial equations and checking that they do not have a common solution there exist many methods for this check thus giving a complete decision procedure for such geometry statements this approach has been recently employed to automatically prove a number of interesting theorems in plane euclidean geometry a grobner basis algorithm is used to check whether a finite set of polynomial equations does not have a solution two other formulations of geometry problems are also discussed and complete methods for solving them using the grobner basis computations are given

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

multifractality of the feigenbaum attractor and fractional derivatives it is shown that fractional derivatives of the integrated invariant measure of the feigenbaum map at the onset of chaos have power law tails in their cumulative distributions whose exponents can be related to the spectrum of singularities f this is a new way of characterizing multifractality in dynamical systems so far applied only to multifractal random functions frisch and matsumoto j stat phys 108 1181 2002 the relation between the thermodynamic approach vul sinai and khanin russian math surveys 39 1 1984 and that based on singularities of the invariant measures is also examined the theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations

quantifier elimination in applied mechanics problems with cylindrical algebraic decomposition abstract the method of cylindrical algebraic decomposition cad of the k dimensional space constitutes a classical technique for the efficient solution of quantifier elimination qe problems in algorithmic computer aided algebra here we apply this method to some applied mechanics problems under appropriate constraints at first we study the problem of a straight elastic beam under a restriction on the maximum permissible deflection along this beam which can easily be reduced to the construction of a one dimensional cad as well as the problem of a circular isotropic elastic medium where a stress component should not exceed a critical value which requires the construction of a three dimensional cad in both these problems we derive also the required quantifier free formulae qffs not including the fundamental variables but only the parameters involved much more difficult cad qff derivation applications concerning an elliptical elastic medium again with an upper bound for a stress component a special case of failure by yielding in fracture mechanics related to sih s strain energy density factor and a frictionless contact problem for an elastic half plane are also considered and explicitly solved with the help of already available cad produced results although evidently cad is not expected to produce qffs in extremely difficult problems finally additional possible applications of cad cqe to applied mechanics problems are also suggested

multifractal structure of clusters and growing aggregates abstract various phenomena on a given fractal object have different critical behavior this is related to the underlying multifractal structure of the aggregate here we discuss the problem with particular emphasis on a scaling approach which leads to a simple real space renormalization group a mechanism which generates multifractality based on a multiplicative process is also illustrated the multifractal structure of the incipient infinite cluster in percolation and diffusion limited aggregation is discussed in detail

growth probability distribution in percolation the authors study the growth probability distribution for a single cluster growth process for percolation at the percolation threshold they find that this distribution is indistinguishable from the chemical path distribution on the percolation cluster as studied by havlin et al 1985 they also study the density profile of the percolation cluster using the multiscaling idea of coniglio and zannetti 1989 they calculate the fractal dimension of the percolation clusters from the density profile and report df 1 895 or 0 005 for two dimensions and 2 50 or 0 01 for three dimensions however they do not find any evidence of multifractal or multiscaling behaviour

invariant radon measures for horocycle flows on abelian covers we classify the ergodic invariant radon measures for horocycle flows on d covers of compact riemannian surfaces of negative curvature thus proving a conjecture of m babillot and f ledrappier an important tool is a result in the ergodic theory of equivalence relations concerning the reduction of the range of a cocycle by the addition of a coboundary