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

fractals in physics squig clusters diffusions fractal measures and the unicity of fractal dimensionality the three topics discussed in this paper are largely independent part 1 fractal squig clusters are introduced and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality physics on these new geometric shapes should prove tractable as background the author s theories of squig intervals and squig trees are reviewed and restated in more versatile form part 2 the notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality scaling noises are touched upon a common formalism is presented for three forms of anomalous diffusion the ant in the fractal labyrinth fractional brownian motion and levy stable motion the fractal dimensionalities common to diverse shapes generated by diffusion are given in table i as functions of the latent dimensionalities of the support of the motion and of the diffusion itself part 3 it is argued that every fractal point set has a unique fractal dimensionality but it is pointed out that many fractals involve diverse combinations of many fractal point sets such is in particular the case for fractal measures and for fractal graphs often called hierarchical lattices the fractal measures that the author had introduced in the early 1970s are described including new developments

coincidence and common fixed and periodic point theorems in cone metric spaces the aim of this paper is to show the existence of coincidence and fixed points for mappings satisfying property e a in cone metric spaces also we give periodic point theorems in cone metric spaces

oscillating singularities on cantor sets a grand canonical multifractal formalism the singular behavior of functions is generally characterized by their holder exponent however we show that this exponent poorly characterizes oscillating singularities we thus introduce a second exponent that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform we then elaborate on a grand canonical multifractal formalism that describes statistically the fluctuations of both the holder and the oscillation exponents we prove that this formalism allows us to recover the generalized singularity spectrum of a large class of fractal functions involving oscillating singularities

multifractal analysis of chinese stock volatilities based on the partition function approach abstract we have performed a detailed multifractal analysis on the 1 min volatility of two indexes and 1139 stocks in the chinese stock markets based on the partition function approach the partition function q s scales as a power law with respect to the box size s the scaling exponents q form a nonlinear function of q statistical tests based on bootstrapping show that the extracted multifractal nature is significant at the 1 significance level the individual securities can be well modeled by the p model in turbulence with p 0 40 0 02 based on the idea of ensemble averaging including quenched and annealed average we treat each stock exchange as a whole and confirm the existence of multifractal nature in the chinese stock markets

regularity criteria for almost every function in sobolev spaces in this paper we determine the multifractal nature of almost every function in the prevalence setting in a given sobolev or besov space according to different regularity exponents these regularity criteria are based on local l p regularity or on wavelet coefficients and give a precise information on pointwise behavior

multifractality in the random parameter model for multivariate time series abstract the random parameter model was proposed to explain the structure of the covariance matrix in problems where most but not all of the eigenvalues of the covariance matrix can be explained by random matrix theory in this article we explore the scaling properties of the model as observed in the multifractal structure of the simulated time series we use the wavelet transform modulus maxima technique to obtain the multifractal spectrum dependence with the parameters of the model the model shows a scaling structure compatible with the stylized facts for a reasonable choice of the parameter values

the bulk of the stock market correlation matrix is not pure noise abstract we analyse the structure of the distribution of eigenvalues of the stock market correlation matrix with increasing length of the time series representing the price changes we use 100 highly capitalized stocks from the american market and relate the result to the corresponding ensemble of wishart random matrices it turns out that systematically more eigenvalues remain beyond the borders prescribed by this variant of the random matrix theory rmt this may indicate that even the bulk of the spectrum of the stock market correlation matrix carries some sort of correlations that are masked by a measurement noise when the time series used to construct the matrix are short we also study some other characteristics of the noisy eigensignals like their return distributions temporal correlations or their multifractal spectra and the results support the above conclusions

common fixed point theorems for contractive maps two common fixed point theorems have been proved by using minimal type commutativity and contractive conditions the last theorem extends known results on compatible maps to a wider class of mappings

golden mean arithmetic in the fractal branching of diffusion limited aggregates we use the wavelet transform microscope to explore the intricate fractal geometry of witten and sander s diffusion limited aggregates we report on the discovery of fibonacci sequences in the internal hierarchical structure of these clusters we discuss the relevance of the golden mean arithmetic to the numerically well established statistical self similarity of these aggregates

locality of turbulent cascades in this paper we establish sufficient conditions for locality of turbulent cascades by an exact analysis of the fluid equations no assumptions of homogeneity or isotropy are invoked and the results apply to individual realizations with no resort to any statistical averaging the only requirement is suitable regularity of the turbulent solutions of the fluid equations in the high reynolds number limit corresponding to holder continuity but non differentiability in space we use a smooth filtering approach to resolve the turbulent fields both in space and in scale we discuss several physical cascades to exemplify and clarify the analysis including joint energy and helicity cascades in three space dimensions and the dual inverse energy cascade and direct enstrophy cascade in two dimensions

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

analysis of singularities from modulus maxima of complex wavelets complex valued wavelets are normally used to measure instantaneous frequencies while real wavelets are normally used to detect singularities we prove that the wavelet modulus maxima with a complex valued wavelet can detect and characterize singularities this is an extension of the previous wavelet work of mallat and hwang on modulus maxima using a real wavelet with this extension we can simultaneously detect instantaneous frequencies and singularities from the wavelet modulus maxima of a complex valued wavelet some results of singularity detection with the modulus maxima from a real wavelet and an analytic complex valued wavelet are compared we also demonstrate that singularity detection methods can be employed to detect the corners of a planar object

integral φ type contractions and existence of continuous solutions for nonlinear integral inclusions abstract in this paper we discuss a general common fixed point theorem of integral type for two pairs of weakly compatible mappings satisfying certain integral type implicit relations in symmetric spaces by using the notion of a pair of mappings satisfying property e a our main result improves and extends the recent results of aliouche a aliouche a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type j math anal appl 322 2006 796 802 pathak tiwari and khan h k pathak rakesh tiwari m s khan a common fixed point theorem satisfying implicit relation appl math e notes 7 2007 222 228 and several other known results subsequently a multi valued version of krasnoselski fixed point theorem in a cone is used to discuss the existence of c 0 t and l p 0 t solutions to nonlinear integral inclusion y t 0 t k t s a s g s y s f s y s y s ds which improves the recent result of agarwal meehan and o regan r p agarwal m meehan d o regan positive l p and continuous solutions for fredholm integral inclusions in set valued mappings with applications in nonlinear analysis gordon breach amsterdam simaa 4 2002 1 9

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

geometrical exponents of contour loops on synthetic multifractal rough surfaces multiplicative hierarchical cascade p model in this paper we study many geometrical properties of contour loops to characterize the morphology of synthetic multifractal rough surfaces which are generated by multiplicative hierarchical cascading processes to this end two different classes of multifractal rough surfaces are numerically simulated as the first group singular measure multifractal rough surfaces are generated by using the p model the smoothened multifractal rough surface then is simulated by convolving the first group with a so called hurst exponent h the generalized multifractal dimension of isoheight lines contours d q correlation exponent of contours x l cumulative distributions of areas and perimeters are calculated for both synthetic multifractal rough surfaces our results show that for both mentioned classes hyperscaling relations for contour loops are the same as that of monofractal systems in contrast to singular measure multifractal rough surfaces h plays a leading role in smoothened multifractal rough surfaces all computed geometrical exponents for the first class depend not only on its hurst exponent but also on the set of p values but in spite of multifractal nature of smoothened surfaces second class the corresponding geometrical exponents are controlled by h the same as what happens for monofractal rough surfaces

equivalent contractive conditions in symmetric spaces in this paper we introduce 22 contractive conditions in symmetric spaces we prove that all these contractive conditions are equivalent then we use these conditions to get some common fixed point theorems for noncompatible maps

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

generalized entropy plane based on permutation entropy and distribution entropy analysis for complex time series abstract entropy is an accessible way to work as a measure of the irregularity and the uncertainty between the predicting knowledge and the given time series statistical complexity measure scm combining shannon entropy and the extensive jensen shannon divergence provides important additional information regarding the peculiarities of the underlying probability distribution not already detected by the entropy in this paper we extend the traditional complexity entropy causality plane which applies the diagram of scm versus normalized shannon entropy to two generalized complexity entropy plane based on permutation entropy pe and permuted distribution entropy pedisen moreover as the important extension of the shannon entropy the tsallis entropy and renyi entropy are used to construct the plane we discuss the parameter selection for the pe plane and pedisen plane respectively outlier detection is recently a heated point focusing on discovering patterns that occur infrequently in the time series in data mining however there exists few entropy plane based methods in outlier detection we apply the proposed procedure to the real world data for outlier detection it turns out that the generalized entropy plane is robust to the type of original series and is efficient for detecting outliers

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

wavelet transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

generalized hurst exponent and multifractal function of original and translated texts mapped into frequency and length time series a nonlinear dynamics approach can be used in order to quantify complexity in written texts as a first step a one dimensional system is examined two written texts by one author lewis carroll are considered together with one translation into an artificial language i e esperanto are mapped into time series their corresponding shuffled versions are used for obtaining a baseline two different one dimensional time series are used here one based on word lengths lts the other on word frequencies fts it is shown that the generalized hurst exponent h q and the derived f curves of the original and translated texts show marked differences the original texts are far from giving a parabolic f function in contrast to the shuffled texts moreover the esperanto text has more extreme values this suggests cascade model like with multiscale time asymmetric features as finally written texts a discussion of the difference and complementarity of mapping into a lts or fts is presented the fts f curves are more opened than the lts ones

improved time series prediction with a new method for selection of model parameters a new method for model selection in prediction of time series is proposed apart from the conventional criterion of minimizing rms error the method also minimizes the error on the distribution of singularities evaluated through the local holder estimates and its probability density spectrum predictions of two simulated and one real time series have been done using kernel principal component regression kpcr and model parameters of kpcr have been selected employing the proposed as well as the conventional method results obtained demonstrate that the proposed method takes into account the sharp changes in a time series and improves the generalization capability of the kpcr model for better prediction of the unseen test data

the necessity of wheels in universal quantization formulas in the context of formal deformation quantization we provide an elementary argument showing that any universal quantization formula necessarily involves graphs with wheels

on the wavelet transformation of fractal objects the wavelet transformation is briefly presented it is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half plane near the boundary of its domain of analyticity as an example a weierstrass like fractal function is considered for which the wavelet transform is related to a jacobi theta function some of the scalings of this theta function are analyzed and give some information about the scaling behavior of this fractal

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

velocity probability density functions of high reynolds number turbulence abstract this paper deals with the probability density function pdf of velocity differences between two points separated by distance r measurements of pdfs were made for r lying in the inertial range for two different flows in a jet with r 852 and in a wind tunnel with r 2720 these pdfs have a characteristic non gaussian shape with exponential tails following kolmogorov s general ideas of log normality a new model for the pdf is developed which contains two parameters determined by experiments this empirical model agrees with the experimental results that the tails of the pdf deviate from a truly exponential behaviour in particular for small r in addition the model leads to the general scaling law ln e 2 r r 0 differenr from kolmogorov s third hypothesis ln e 2 ln r r 0 restricted to the inertial range only x is x x we develop also a formalism based on an extremum principle which is consistent with both the log normality of e and the above mentioned power law in this formalism can be interpreted as the codimension of dissipative structures and asymptotically varies as 1 ln r

invasion percolation and statistics of us treasury bonds abstract using a model of analytic branching cascades on fractal clusters it is shown that us treasury securities returns fluctuations exhibit three types of stochastic behaviors a quasi brownian type for a very small cluster consisting of just a single bond b lognormal type for a medium size very compact cluster branching dimension d 1 and c a stretched lognormal distribution corresponding to branching dimension d 2 for the largest cluster an analogy with multifractality at the onset of chaos is also discussed

common fixed points under strict contractive conditions abstract the aim of the present paper is to obtain common fixed point theorems under strict contractive conditions by using a minimal commutativity condition our results extend the recent results due to pant 1998 j math anal appl 226 251 258 1999 soochow j math 25 37 42

inverse problems in fractal construction moment method solution abstract the inverse problem for several classes of fractals in one and two dimensions can be profitably solved via the application of the classical theory of moments this new perspective leads to important results on the reconstruction process when only a finite number of moments of the fractal measures are known while discussing the rigorous aspects of the theory we also implement it via efficient numerical algorithms

wavelet transform based multifractal formalism in outlier detection and localisation for financial time series we present a method of detecting and localising outliers in financial time series and other stochastic processes the method checks the internal consistency of the scaling behaviour of the process within the paradigm of the multifractal spectrum deviation from the expected spectrum is interpreted as the potential presence of outliers the detection part of the method is then supplemented by the localisation analysis part using the local scaling properties of the time series localised outliers can then be removed one by one with the possibility of dynamic verification of spectral properties both the multifractal spectrum formalism and the local scaling properties of the time series are implemented on the wavelet transform modulus maxima tree

wavelet based estimators of scaling behavior various wavelet based estimators of self similarity or long range dependence scaling exponent are studied extensively these estimators mainly include the bi orthogonal wavelet estimators and the wavelet transform modulus maxima wtmm estimator this study focuses both on short and long time series in the framework of fractional autoregressive integrated moving average farima processes we advocate the use of approximately adapted wavelet estimators for these ideal processes the scaling behavior actually extends down to the smallest scale i e the sampling period of the time series if an adapted decomposition is used but in practical situations there generally exists a cutoff scale below which the scaling behavior no longer holds we test the robustness of the set of wavelet based estimators with respect to that cutoff scale as well as to the specific density of the underlying law of the process in all situations the wtmm estimator is shown to be the best or among the best estimators in terms of the mean squared error mse we also compare the wavelet estimators with the detrended fluctuation analysis dfa estimator which was previously proved to be among the best estimators which are not wavelet based estimators the wtmm estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior

multifractal formalism by enforcing the universal behavior of scaling functions a b s t r a c t despite its solid foundations multifractal analysis is still a challenging task the inversed singularity spectrum is a major pitfall in standard multifractal analyses especially for empirical signals to resolve this issue we identified the fan like convergent geometry of scaling functions yielding a limit value termed focus for all moments at the largest scale building on this behavior of scaling we introduce the novel concept of focus based multifractal formalism it relies on enforcing this universal behavior when the moment wise scaling exponents are assessed for the scaling functions besides developing focus based variants of the established multifractal detrended fluctuation analysis and the wavelet leader method we present a novel analytical tool of multifractal signal summation conversion all methods are extensively tested on exact multifractal signals synthesized by the generalized binomial multifractal model in terms of precision and incidence of inversed singularity spectra our focus based variants never yielded inversed spectra and their precision was found similar to that of standard methods our approach allowed computing a moment wise and a global error parameter describing the impact of finite size effect and degree of multifractality as compared to that of the fitted exact multifractal model we demonstrate that the standard approach to multifractal analyses contains a central element that is essentially monofractal due to its regression scheme assessing the scaling exponents for each and every moment separately hence these methods can yield reliable estimates only for ideally behaving multifractal signals in contrast our focus based variants due to their genuine multifractal model fitting always yield reliable estimates accompanied by goodness of fit statistics the presented novel multifractal tools offer means of dealing with the consequences of endogenous impurities of potentially multifractal empirical signals

limit lognormal multifractal as an exponential functional the limiting distribution of the limit lognormal multifractal first introduced by mandelbrot statistical models and turbulence m rosenblatt and c van atta eds lecture notes in physics 12 springer new york 1972 p 333 and constructed explicitly by bacry et al phys rev e 64 026103 2001 is investigated using its laplace transform a partial differential equation for the laplace transform is derived and it is shown that multifractality alone does not determine the limiting distribution the increments of the limit multifractal process are strongly stochastically dependent the precise nature of this stochastic dependence structure of increments sdsi is the determining characteristic of the limiting distribution the sdsi of the limit process is quantified by means of two integro differential relations obtained by renormalization in the sense of leipnik j aust math soc b 32 327 347 1991 one is interpreted as a counterpart of the star equation of mandelbrot and the other is shown to be an analogue of the classical girsanov theorem in the weak intermittency limit an approximate single variable equation for the laplace transform is obtained and successfully tested numerically by simulation

the dimension spectrum of some dynamical systems we analyze the dimension spectrum previously introduced and measured experimentally by jensen kadanoff and libchaber using large deviation theory we prove for some invariant measures of expanding markov maps that the hausdorff dimensionf of the set on which the measure has a singularity is a well defined concave and regular function in particular we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number we also show in these particular cases that the functionf is universal

multifractal formalism for fractal signals the structure function approach versus the wavelet transform modulus maxima method several attempts have been made recently to generalize the multifractal formalism originally introduced for singular measures to fractal signals we report on a systematic comparison between the structure function approach pioneered by parisi and frisch in 2 proceedings of the international school on turbulence and predictability in geophysical fluid dynamics and climate dynamics edited by m ghil r benzi and g parisi north holland amsterdam 1985 p 84 to account for the multifractal nature of fully developed turbulent signals and an alternative method we have developed within the framework of the wavelet transform analysis we comment on the intrinsic limitations of the structure function approach this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases we demonstrate that our method based on the wavelet transform modulus maxima representation works in most situations and is likely to be the ground of a unified multifractal description of self affine distributions our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations

advanced topology on the multiscale sequence spaces s nu abstract we pursue the study of the multiscale spaces s introduced by jaffard in the context of multifractal analysis we give the necessary and sufficient condition for s to be locally p convex and exhibit a sequence of p norms that defines its natural topology the strong topological dual of s is identified to another sequence space depending on endowed with an inductive limit topology as a particular case we describe the dual of a countable intersection of besov spaces

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

singularity spectrum of fractal signals from wavelet analysis exact results the multifractal formalism for singular measures is revisited using the wavelet transform for bernoulli invariant measures of some expanding markov maps the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima the generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures it is demonstrated that the hausdorff dimensiond h of the set of singularities of holder exponenth can be directly determined from the wavelet transform modulus maxima the singularity spectrum so obtained is shown to be not disturbed by the presence in the signal of a superimposed polynomial behavior of ordern provided one uses an analyzing wavelet that possesses at leastn n vanishing moments however it is shown that ac behavior generally induces a phase transition in thed h singularity spectrum that somewhat masks the weakest singularities this phase transition actually depends on the numbern of vanishing moments of the analyzing wavelet its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal these theoretical results are illustrated with numerical examples they are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional brownian motions and turbulent velocity signals

edges transitions and criticality in this article various notions of edges encountered in digital image processing are reviewed in terms of compact representation or completion we show that critical exponents defined in statistical physics lead to a much more coherent definition of edges consistent across the scales in acquisitions of natural phenomena such as high resolution natural images or turbulent acquisitions edges belong to the multiscale hierarchy of an underlying dynamics they are understood from a statistical perspective well adapted to fit the case of natural images numerical computation methods for the evaluation of critical exponents in the non ergodic case are recalled which apply for the vast majority of natural images we study the framework of reconstructible systems in a microcanonical formulation show how it redefines edge completion and how it can be used to evaluate and assess quantitatively the adequation of edges as candidates for compact representations we study with particular attention the case of turbulent data in which edges in the classical sense are particularly challenged tests are conducted and evaluated on a standard database for natural images we test the newly introduced compact representation as an ideal candidate for evaluating turbulent cascading properties of complex images and we show better reconstruction performance than the classical tested methods highlightsclassical edge detection algorithms are reviewed in terms of completion a new edge detection scheme based on multiscale hierarchical classification assessing the consistency of edges across scales evaluating edge performance through the framework of reconstructible systems comparing the performance with classical edge operators

coincidence and fixed points in symmetric spaces under strict contractions abstract some common fixed point theorems due to aamri and el moutawakil m aamri d el moutawakil some new common fixed point theorems under strict contractive conditions j math anal appl 270 2002 181 188 and pant and pant r p pant v pant common fixed points under strict contractive conditions j math anal appl 248 2000 327 332 proved for strict contractive mappings in metric spaces are extended to symmetric semi metric spaces under tight conditions some related results are derived besides discussing illustrative examples which establish the utility of results proved in this note

a path integral approach to the kontsevich quantization formula we give a quantum field theory interpretation of kontsevich s deformation quantization formula for poisson manifolds we show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory its batalin vilkovisky quantization yields a superconformal field theory the associativity of the star product and more generally the formality conjecture can then be understood by field theory methods as an application we compute the center of the deformed algebra in terms of the center of the poisson algebra

wavelet analysis of a gaussian kolmogorov signal several attempts have been made recently to use wavelets transforms for extracting histograms of scaling exponents from experimental turbulence data similar techniques are here applied to a gaussian signal having a kolmogorov 5 3 energy spectrum this is an instance of the class of fractional brownian motions having scaling exponent 1 3 almost everywhere for the gaussian signal a spurious non trivial histogram is obtained by applying a mexican hat wavelet transform analysis on the other hand we show that the use of complex wavelets and even more so the application of an optimal wavelet transform method strongly reduce the spurious fluctuations observed in the processing of the gaussian signal

testing for the source of multifractality in water level records abstract the multifractality in hydrologic data has been studied in many literatures but few literatures focus on its source which is helpful for understanding the underlying mechanisms that generate hydrologic data we propose a hypothesis testing procedure for the source of multifractality in water level records based on the empirical distributions of generalized hurst exponent estimated from a set of shuffled or surrogate series the proposed hypothesis testing procedure can show more details about the source of multifractality than previous methods with some statistics especially about the effects of large and small fluctuations on multifractality the generalized hurst exponents are estimated via multifractal detrended fluctuation analysis the data set contains about two million high frequency water level records of a northern china river at its 10 observation stations the testing results show that the multifractality in water levels is mainly caused by nonlinear correlations in small fluctuations and linear correlations in small and large fluctuations and is also related to the probability distribution of small fluctuations this conclusion is validated via some constructed semi surrogate series and values of statistics for source testing

low order variability diagrams for short range correlation evidence in financial data bgl usd exchange rate dow jones industrial average gold ounce price a method to sort out short range correlations and decorrelations in financial data is tested on three typical sets the bulgarian lev usa dollar bgl usd exchange rate the dow jones industrial average the gold ounce price the method makes use of the so called variability diagram technique three toys are used as models in order to understand features our findings indicate that some predictability can be found at short range time intervals c 1999 elsevier science b v all rights reserved

wavelet analysis of fractal boundaries part 1 local exponents let be a domain of rd in part 1 of this paper we introduce new tools in order to analyse the local behavior of the boundary of classifica tions based on geometric accessibility conditions are introduced and compared they are related to analytic criteria based either on local lp regularity of the characteristic function 1 or on its wavelet coefficients part 2 deals with the global analysis of the boundary of we develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur these methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

deformation quantization of polynomial poisson algebras abstract this paper discusses the notion of a deformation quantization for an arbitrary polynomial poisson algebra a we compute an explicit third order deformation quantization of a and show that it comes from a quantized enveloping algebra we show that this deformation extends to a fourth order deformation if and only if the quantized enveloping algebra gives a fourth order deformation moreover we give an example where the deformation does not extend a correction term to the third order quantization given by the enveloping algebra is computed which precisely cancels the obstruction so that the modified third order deformation extends to a fourth order one the solution is generically unique up to equivalence

long term correlations and multifractal nature in the intertrade durations of a liquid chinese stock and its warrant abstract the intertrade duration of equities is an important financial measure characterizing trading activities it is defined as the waiting time between successive trades of an equity using the ultrahigh frequency data of a liquid chinese stock and its associated warrant we perform a comparative investigation of the statistical properties of their intertrade duration time series the distributions of the two equities can be better described by the shifted power law form than the weibull form and their scaled distributions do not collapse onto a single curve although the intertrade durations of the two equities have very different magnitude their intraday patterns exhibit very similar shapes both detrended fluctuation analysis dfa and detrending moving average analysis dma show that the 1 min intertrade duration time series of the two equities are strongly correlated in addition both multifractal detrended fluctuation analysis mfdfa and multifractal detrending moving average analysis mfdma unveil that the 1 min intertrade durations possess multifractal nature however the difference between the two singularity spectra of the two equities obtained from the mfdma is much smaller than that from the mfdfa

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

scaling characteristics in aftershock sequence of earthquake abstract the possible scale invariant behavior and the clustering characteristics in aftershock sequence of chi chi taiwan main earthquake asccme that occurred in 1999 9 20 17 47 were investigated by means of some statistical tools histogram spectral analysis and fractal theory the examined data were constructed from the aftershocks that occurred at the locations defined at longitude 120 1 121 3 and latitude 23 3 24 5 during the 1999 9 20 17 47 1999 9 24 08 13 period it was found that the aftershock sequence exhibited the characteristic right skewed frequency distribution and could be well described with the lognormal distribution long term memory and the possibility of scale invariance were first roughly identified through the analysis of autocorrelation and power spectrum respectively scale invariance was clearly revealed with the aid of box counting method and the box dimension was shown to be a decreasing function of the threshold magnitude level i e the weak and intense regions scaled differently to verify the existence of multifractal characteristics the aftershock sequence was transferred into a useful compact form through the multifractal formalism namely the q q and f plots the analysis confirmed the existence of multifractal characteristics in the examined aftershock sequence the origin of both the pronounced right skewness and multifractal phenomena in aftershock sequence might be interpreted in terms of the multiplicative cascade process of the stress in the earth s crust a simple two scale cantor set with unequal scales and weights was then used to fit the calculated q q plot this model fitted remarkably well the entire spectrum of scaling exponents of the examined asccme

some new common fixed point theorems under strict contractive conditions the main purpose of this paper is to give some new common fixed point theorems under strict contractive conditions for mappings satisfying a new property

three dimensional wavelet based multifractal method the need for revisiting the multifractal description of turbulence dissipation data we generalize the wavelet transform modulus maxima wtmm method to multifractal analysis of 3d random fields this method is calibrated on synthetic 3d monofractal fractional brownian fields and on 3d multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration then we apply the 3d wtmm method to the dissipation field issued from 3d isotropic turbulence simulations we comment on the need to revisit previous box counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures

structural five fold symmetry in the fractal morphology of diffusion limited aggregates abstract the statistical self similarity of the geometry of diffusion limited aggregates and the multifractal nature of the growth probability distribution on the surface of the growing clusters are investigated using the wavelet transform this study reveals the existence of a predominant structural five fold symmetry in the internal frozen region as well as in the active outer region of the interface this observation is corroborated by a statistical analysis of the screening effects that govern diffusion limited aggregation dla growth in linear and sector shaped cells the existence of this symmetry is likely to be a clue to a hierarchichal fractal ordering we report on the discovery of fibonacci sequences in the inner extinct region of large mass off lattice dla clusters with a branching ratio which converges asymptotically to the golden mean we suggest an interpretation of the dla morphology as a quasifractal counterpart of the well ordered snowflake fractal architecture

complex temperature singularities and their application to the thermodynamic description of multifractals it is shown that applicability of standard high temperature expansions in multifractal thermodynamics is restricted by complex temperature singularities an analytic continuation method finite temperature expansions has been developed to improve the analytic expansions approach the quantum intermittency is then used as an example of applicability of the finite temperature expansion

multifractal estimates of monofractality in rr heart series in power spectrum ranges abstract two popular estimators of multifractal properties the wavelet transform modulus maxima method and multifractal detrended fluctuation analysis are applied to investigate signals consisting of normal rr series in 39 healthy subjects and 90 patients suffering from systolic dysfunction of the left ventricle however differently from standards for multifractal analysis the scaling is performed separately in intervals corresponding to standard power spectral bands low lf very low vlf and ultra low frequencies ulf tests on fractional brownian motions fbm are done to quantify properties of the estimators as detectors of monofractality in lf vlf and ulf bands arguments are given that multifractal analysis of rr series performed in these bands has a physiological meaning the increased activation of the sympathetic nervous system caused by heart disease is detected evidently only by analysis in lf the transition in multifractal characteristics between diurnal and nocturnal activity takes place when the analysis moves from lf and vlf to ulf only in ulf the diurnal heart rate variability can be approximated by fbm with a self similarity index of h 0 20

bifractality of human dna strand asymmetry profiles results from transcription we use the wavelet transform modulus maxima method to investigate the multifractal properties of strand asymmetry dna walk profiles in the human genome this study reveals the bifractal nature of these profiles which involve two competing scale invariant up to repeat masked distances ensuremath lesssim 40 phantom rule 0 3em 0ex mathrm kbp components characterized by h older exponents h 1 0 78 and h 2 1 respectively the former corresponds to the long range correlated homogeneous fluctuations previously observed in dna walks generated with structural codings the latter is associated with the presence of jumps in the original strand asymmetry noisy signal s we show that a majority of upward downward jumps colocate with gene transcription start end sites here 7228 human gene transcription start sites from the refgene database are found within 2 phantom rule 0 3em 0ex mathrm kbp from an upward jump of amplitude ensuremath delta s ensuremath geqslant 0 1 which suggests that about 36 of annotated human genes present significant transcription induced strand asymmetry and very likely high expression rate