on non equilibrium stochastic dynamics for interacting particle systems in continuum abstract we propose a general scheme for construction of markov stochastic dynamics on configuration spaces in continuum an application to the glauber type dynamics with competitions is considered

efficient computation of the hutchinson metric between digitized images the hutchinson metric is a natural measure of the discrepancy between two images for use in fractal image processing an efficient solution to the problem of computing the hutchinson metric between two arbitrary digitized images is considered the technique proposed here based on the shape of the objects as projected on the digitized screen can be used as an effective way to establish the error between the original and the possibly compressed decoded image to test the performance of our method we apply it to compare pairs of fractal objects as well as to compare real world images with the corresponding reconstructed ones

robust estimation of parameter for fractal inverse problem in this paper some similarity measures for fractal image compression fic are introduced which are robust against noises in the proposed methods robust estimation technique from statistics is embedded into the encoding procedure of the fractal inverse problem to find the parameters when the original image is corrupted by noises we hope that the proposed scheme is insensitive to those noises presented in the corrupted image this leads to a new concept of robust estimation of fractal inverse problem the proposed least absolute derivation lad least trimmed squares lts and wilcoxon fic are the first attempt toward the design of robust fractal image compression which can remove the noises in the encoding process the main disadvantage of the robust fic is the computational cost to overcome this drawback particle swarm optimization pso technique is utilized to reduce the searching time simulation results show that the proposed fic is robust against the outliers in the image also the pso method can effectively reduce the encoding time while retaining the quality of the retrieved image

analytic expression for the structure factor and for the moment generating function of fractal sets and multifractal measures by applying the definition of markov operator for iterated function systems with probability an analytic expression for the structure factor and for the moment generating function of a class of fractal sets is obtained in the presence of arbitrary multifractal distributions associated with affine iterated function systems the properties of the structure factor of fractal sets are analysed in detail the results developed for the structure factors have interesting implications as regards to the theory of laplace transforms of singular non atomic measures some physical applications of the theory related to fractal and heterogeneous systems in the case of adsorption and relaxation are briefly addressed integral transform theory is also applied to solve the dirichlet problem for the laplace equation on the circle in the presence of singular non atomic boundary conditions

duality in parameter space and approximation of measures for mixing repellers for one dimensional expanding mapst with an invariant measure we consider in a parameter space the envelope n of the real lines associated to any couple of points of the orbit connected byn iterations oft if the map hass inverses and is piecewise linear then the sets n are just the union ofs n points and converge to the invariant cantor set oft a correspondence between all the sets and their measures is established and allows one to associate the atomic measure on 1 to the completly continuous measure on the cantor set if the map is nonlinear hyperbolic and hass inverses the sets n are homeomorphic to the cantor set they converge to the cantor set oft and their measures converge to the measure of the cantor set whenn the correspondence between the sets n allows one to define converging approximation schemes for the map an its measure one replaces each of thes n disjoint sets with a point in a convenient neighborhood and a probability equal to its measure and transforms it back to the original set 1 all the approximations with linear cantor systems previously proposed are recovered the converging proprties being straightforward in the present scheme moreover extensions to higher dimensionality and to nondisconnected repellers arte possible and are briefly examined

diffusion on multifractals abstract we introduce a class of random fields with variable mean square regularity order defined on multifractal domains the local singularity order of these random fields then depends on the initial variable mean square regularity order and on the variable local singularity exponent of the multifractal measure defining the local dimension of the domain considered the theory is developed in a generalized framework through the covariance factorization using the tools of reproducing kernel hilbert spaces and fractional sobolev spaces of variable order

stochastic perturbation of dynamical systems the weak convergence of measures in this paper we consider the asymptotic behaviour of randomly perturbed discrete dynamical systems we treat this problem by examining the evolution of the corresponding sequences of distributions we show that an average contractive property is sufficient to ensure the weak convergence of the sequence of distributions to a unique stationary measure

multifractal decompositions of moran fractals we present a rigorous construction and generalization of the multifractal decomposition for moran fractals with infinite product measure the generalization is specified by a system of nonnegative weights in the partition sum all the usual smooth properties of the f theory are recovered for the case that the weights are equal to unity the generalized spectrum f w is invariant to a group of gauge transformations of the weights and in addition need no longer be concave in case the fractal is a cantor set generated by an iterated function system of similarities is the pointwise dimension of the measure we discuss properties of some examples

an improved fractal image compression approach by using iterated function system and genetic algorithm abstract this paper presents an improved method of generating a binary image affine ifs iterated function system by using genetic algorithm we adopt a natural variable length genotype encoding to represent an individual the multiobject fitness function is also applied in this algorithm in addition a distributed version of the binary image compression algorithm is implemented both theoretical analysis and experimental results show a higher compression ratio with better quality images by using the proposed algorithm

fractal image denoising over the past decade there has been significant interest in fractal coding for the purpose of image compression however applications of fractal based coding to other aspects of image processing have received little attention we propose a fractal based method to enhance and restore a noisy image if the noisy image is simply fractally coded a significant amount of the noise is suppressed however one can go a step further and estimate the fractal code of the original noise free image from that of the noisy image based upon a knowledge or estimate of the variance of the noise assumed to be zero mean stationary and gaussian the resulting fractal code yields a significantly enhanced and restored representation of the original noisy image the enhancement is consistent with the human visual system where extra smoothing is performed in flat and low activity regions and a lower degree of smoothing is performed near high frequency components e g edges of the image we find that for significant noise variance spl sigma spl ges 20 the fractal based scheme yields results that are generally better than those obtained by the lee filter which uses a localized first order filtering process similar to fractal schemes we also show that the lee filter and the fractal method are closely related

huber fractal image coding based on a fitting plane recently there has been significant interest in robust fractal image coding for the purpose of robustness against outliers however the known robust fractal coding methods hfic and lad fic etc are not optimal since besides the high computational cost they use the corrupted domain block as the independent variable in the robust regression model which may adversely affect the robust estimator to calculate the fractal parameters depending on the noise level this paper presents a huber fitting plane based fractal image coding hfpfic method this method builds huber fitting planes hfps for the domain and range blocks respectively ensuring the use of an uncorrupted independent variable in the robust model on this basis a new matching error function is introduced to robustly evaluate the best scaling factor meanwhile a median absolute deviation mad about the median decomposition criterion is proposed to achieve fast adaptive quadtree partitioning for the image corrupted by salt pepper noise in order to reduce computational cost the no search method is applied to speedup the encoding process experimental results show that the proposed hfpfic can yield superior performance over conventional robust fractal image coding methods in encoding speed and the quality of the restored image furthermore the no search method can significantly reduce encoding time and achieve less than 2 0 s for the hfpfic with acceptable image quality degradation in addition we show that combined with the mad decomposition scheme the hfp technique used as a robust method can further reduce the encoding time while maintaining image quality

iterated fuzzy set systems a new approach to the inverse problem for fractals and other sets abstract images with grey or colour levels admit a natural representation in terms of fuzzy sets but without the usual probabilistic interpretation of the latter we introduce a fuzzy set approach which incorporates in part the technique of iterated function systems ifs for the construction analysis and or approximation of typically fractal sets and images the method represents a significant departure from ifs especially in the interpretation of the resulting image the introduction of grey level maps i 0 1 0 1 associated with the contractive maps w i of the ifs affords much greater flexibility in the generation of images as well as in the inverse problem

extreme value laws for fractal intensity functions in dynamical systems minkowski analysis typically in the dynamical theory of extremal events the function that gauges the intensity of a phenomenon is assumed to be convex and maximal or singular at a single or at most a finite collection of points in phase space in this paper we generalize this situation to fractal landscapes i e intensity functions characterized by an uncountable set of singularities located on a cantor set this reveals the dynamical role of classical quantities like the minkowski dimension and content whose definition we extend to account for singular continuous invariant measures we also introduce the concept of extremely rare event quantified by non standard minkowski constants and we study its consequences to extreme value statistics limit laws are derived from formal calculations and are verified by numerical experiments

approximations of fractal sets abstract five approximations of fractal sets a are compared the first three were described by hutchinson barnsley demko and williams respectively two new other approximations are still efficient in all circumstances for any algorithm error estimates in the approximations are found numerical comparisons are also done through classical examples of attractors in the plane

recognition of an organism from fragments of its complete genome this paper considers the problem of matching a fragment to an organism using its complete genome our method is based on the probability measure representation of a genome we first demonstrate that these probability measures can be modeled as recurrent iterated function systems rifs consisting of four contractive similarities our hypothesis is that the multifractal characteristics of the probability measure of a complete genome as captured by the rifs is preserved in its reasonably long fragments we compute the rifs of fragments of various lengths and random starting points and compare with that of the original sequence for recognition using the euclidean distance a demonstration on five randomly selected organisms supports the above hypothesis

a chaos game algorithm for generalized iterated function systems in this paper we provide an extension of the classical chaos game for ifsp the paper is divided into two parts in the first one we discuss how to determine the integral with respect to a measure which is a combination of a self similar measure from an ifsp along with a density given by an ifsm in the second part we prove a version of the ergodic theorem for the integration of a continuous multifunction with respect to the invariant measure of an ifsp these results are in line with some recent extensions of ifs theory to multifunctions

vlasov scaling for stochastic dynamics of continuous systems we describe a general derivation scheme for the vlasov type equations for markov evolutions of particle systems in continuum this scheme is based on a proper scaling of corresponding markov generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations several examples of realization of the proposed approach in particular models are presented

recurrent random walks in nonnegative matrices ii in this paper we continue the study undertaken in our earlier paper m1 one of the main results here can be described as follows letx0 x1 be a sequence of iid random affine maps from r d into itself let us write w n x n x n 1 x0 andz n x0x1 x n where composition of maps is the rule of multiplication by the attractora u u r d we mean the setau y r d p wnu n i o 0 for every openn containingy it is shown that the attractora u under mild conditions is the support of a stationary probability measure when the random walk z n has at least one recurrent state

the dimension of affine invariant fractals a generic formula for the hausdorff and box counting dimensions of self affine fractals is presented extensions to the non linear situation and to repellers in discrete dynamical systems are discussed

calculating derivatives with respect to parameters in iterated function systems abstract an iterated function system is a dynamical system consisting of several contracting maps with associated probabilities under certain general assumptions long term average values of physical quantities associated with an iterated function system which is dependent on an external parameter are differentiable functions of the parameter we present methods for the calculation of the derivatives of average values with respect to a parameter these algorithms yield values for these derivatives with an amount of computation proportional to the calculation of the average values themselves to the same accuracy we also show how the derivatives might be calculated in the case where the probabilities vary with position and with the parameter

mesh generation refinement using fractal concepts and iterated function systems a novel method of mesh generation is proposed which is based on the use of fractal concepts to derive contractive affine transformations the transformations are constructed in such a manner that the attractors of the resulting maps are a union of the points lines and surfaces in the domain in particular the mesh nodes may be generated recursively as a sequence of points which are obtained by applying the transformations to a coarse background mesh constructed from the given boundary data a delaunay triangulation or similar edge connection approach can then be performed on the resulting set of nodes in order to generate the mesh local refinement of an existing mesh can also be performed using the procedure the method is easily extended to three dimensions in which case the delaunay triangulation is replaced by an analogous 3 d tesselation

regulation mechanisms in spatial stochastic development models the aim of this paper is to analyze different regulation mechanisms in spatial continuous stochastic development models we describe the density behavior for models with global mortality and local establishment rates we prove that the local self regulation via a competition mechanism density dependent mortality may suppress a unbounded growth of the averaged density if the competition kernel is superstable

some properties of mixing repellers applying the thermodynamic formalism to mixing repellers for regular maps the author obtains some rigorous relations between the dimensions of the repellers and the dynamic properties of the transformation on them for a class of one dimensional expanding maps these quantities can be approximated to any desired accuracy by means of the corresponding variables for linear expanding maps which can be computed exactly finally a mathematical foundation and some properties of the generalised dimensions are given

global analysis of self similarity features of cellular automata selected examples abstract the analysis of self similarity properties of the evolution patterns of linear cellular automata is investigated using the theory of hierarchical iterated function systems and substitution systems we present several selected examples

on computing jacobi matrices associated with recurrent and mobius iterated function systems a new technique for the computation of jacobi matrices associated with measures possessing scale invariance properties is described invariance is imposed via linear transformations and by mobius transformations in psl 2 r this technique is proven to be numerically stable and apt to compute jacobi matrices of large orders

chaotic attractors of a locally conservative hyperbolic map with overlap abstract we introduce a class of hyperbolic maps which are locally conservative but globally dissipative the dissipation stems not from local contraction of phase space but from the overlap that arises from the noninvertibility of the map the dynamics of these maps are chaotic and the attractors are strange although not necessarily fractal though several examples of this class are provided our analysis concentrates on a single example which is obtained as a modification of the baker s transform we discuss the morphology of the attractors as the map parameter is varied we find that some attractors are thin fractals and others are fat sets of positive measure we are able to characterize the parameter values that lead to these two kinds of attractors and for certain parameter values we can derive analytically exact values for the fractal dimension or for the fat fractal scaling exponent taken together these results lead us to conjecture that the generic attractor of this map has zero measure and full dimension

and the rolling ball 2014 john c baez and john huerta understanding the exceptional lie groups as the symmetry groups of simpler objects is a long standing program in mathematics here we explore one famous realization of the smallest exceptional lie group g2 its lie algebra g2acts locally as the symmetries of a ball rolling on a larger ball but only when the ratio of radii is 1 3 using the split octonions we devise a similar but more global picture of g2 it acts as the symmetries of a spinorial ball rolling on a projective plane again when the ratio of radii is 1 3 we explain this ratio in simple terms use the dot product and cross product of split octonions to describe the g2incidence geometry and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations

normal forms and versal deformations of linear hamiltonian systems abstract equivalence classes of time independent linear real hamiltonian systems can be identified up to canonical transformations with the orbits of the adjoint action of the real symplectic group on its lie algebra a new set of representatives also called normal forms for these orbits is given versal deformations of systems in normal form are constructed applications of versal deformations to the study of bifurcations of linear systems with small codimension are indicated

versal normal form for nonsemisimple singularities abstract the theory of versal normal form has been playing a role in normal form since the introduction of the concept by v i arnol d in 1 2 but there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields that is the linear part should be done completely first before one computes the nonlinear terms in this paper we address this issue by giving a complete description of a first order calculation in the case of the two and three dimensional irreducible nilpotent cases which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics l 4 problem

meromorphic structure of the mellin transforms and short distance behavior of correlation integrals the short distance behavior of the measure of a sphere and of the correlation integral is determined in the case of disconnected repellers by scaling laws whose corrections are oscillating functions periodic or aperiodic depending on exact or approximate self similarity of the measure the mellin transforms prove to be the correct analytic tool in order to investigate these corrections to scaling it has been previously proved that they are meromorphic for linear cantor sets and that the leading pole gives the correlation dimension in agreement with the results of the thermodynamic formalism here we show that the residues of these poles can also be computed to any desired accuracy with simple algorithms and that the knowledge of the singularity spectrum of the mellin transforms provides the fourier spectrum of the scaling correction for the self similar measure and that it reproduces the damped oscillations in the generic case the method applies to the nonlinear repellers such as the disconnected julia sets by using an approximation theorem

refinable functions functionals and iterated function systems invariant measures of iterated function systems and refinable functions are mathematical tools of vast usage that although usually considered in different contexts are deeply linked we outline these links in particular with respect to existence and regularity of these objects

iterated function systems a rising from recursive estimation problems we show that under suitable conditions the one step predictor of a finite state markov chain from noisy observations has a unique stationary law which is supported by a self similar set called the attractor under additional symmetry conditions such attractor is either connected or totally disconnected and perfect in this latter case the predictor keeps an infinite memory of the past observations the main problem of interest is to identify those values of the parameters of the chain and the observation process for which this happens in the binary case the problem is completely solved in higher dimension the problem is harder a complete solution is presented for ternary chains in the completely symmetric persistent case

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

missing moment and perturbative methods for polynomial iterated function systems abstract an iterated function system ifs over a compact metric space x is defined by a set of contractive maps wi x x i 1 n with associated nonzero probabilities pi 0 p i 1 the parallel action of the maps defines a unique compact invariant attractor set a x which supports an invariant measure and which is balanced with respect to the pi for linear w i on x r the invariance of yields a relation between the moments gn nd which permits their recursive computation from the initial value g0 1 for nonlinearwi however the moment relations are incomplete and do not permit a recursive computation this paper describes two methods of obtaining accurate estimates of the moments when the ifs maps wi are polynomials i application of the necessary hausdorff conditions on the gi to obtain convergent upper and lower bounds and ii a perturbation expansion approach the methods are applied to some model problems

markov chains in random environments and random iterated function systems we consider random iterated function systems giving rise to markov chains in random stationary environments conditions ensuring unique ergodicity and a pure type characterization of the limiting randomly invariant probability measure are provided we also give a dimension formula and an algorithm for simulating exact samples from the limiting probability measure

on local fractal functions in besov and triebel lizorkin spaces abstract within the new concept of a local iterated function system local ifs we consider a class of attractors of such ifss namely those that are graphs of functions these new functions are called local fractal functions and they extend and generalize those that are currently found in the fractal literature for a class of local fractal functions we derive explicit conditions for them to be elements of besov and triebel lizorkin spaces these two scales of functions spaces play an important role in interpolation theory and for certain ranges of their defining parameters describe many classical function spaces in the sense of equivalent norms the conditions we derive provide immediate information about inclusion of local fractal functions in for instance lebesgue sobolev slobodeckij holder bessel potential and local hardy spaces

a projective algorithm for approximation of fractal sets a new projective algorithm on approximation of fractal sets is described in this paper compared with five other algorithms the projective algorithm is efficient in all circumstances error estimates are obtained in the approximation process in this algorithm numerical comparisons are done through classical examples of attractors in the plane

iterated function systems on multifunctions and inverse problems abstract in this paper we first consider the problem of defining ifs operators on the space k c of non empty compact and convex subsets of r d after defining a complete metric on k c we construct an ifs operator and show some properties a notable feature is the definition of a type of weak inner product on k c we then define a family of complete metrics on the space of all measurable set valued functions with values in k c and extend the weak inner product to this space following this we construct ifs operators on these spaces we close with a brief discussion of the inverse problem of approximating an arbitrary multifunction by the attractor of an ifs

fractal analysis of measure representation of large proteins based on the detailed hp model abstract the notion of measure representation of protein sequences is introduced based on the detailed hp model multifractal analysis and detrended fluctuation analysis are then performed on the measure representations of a large number of long protein sequences it is concluded that these protein sequences are not completely random sequences through the measure representations and the values of the d q spectra and related c q curves the values of the exponent from the detrended fluctuation analysis show that the k strings with the ordering in the measure representation exhibit strong long range correlation for substrings with length k 5 the d q spectra of all proteins studied are multifractal like and sufficiently smooth for the c q curves to be meaningful the c q curves of all proteins resemble a classical phase transition at a critical point an ifs model is found to simulate the measure representation of protein sequences very well from the estimated values of parameters in the ifs model we think the non polar residues and uncharged polar residues play a more important role than other kinds of residues in the protein folding process

fractal block coding a functional approach for image and signal processing in this paper we show how the generalized self similarity model introduced by cabrelli et al in 1 can be used for the coding and reconstruction of digital signals and images we also prove how the block coding techniques introduced by jacquin 2 for measures flt naturally into this functional model allowing us to take advantage of local redundancy of the images as well as to fully automate the encoding and decoding process c

stable recovery of fractal measures by polynomial sampling abstract the problem of using iterated function systems ifs to approximate fractal measures from polynomial samples is studied a method based on generalized moments is proposed this method does not suffer from the ill conditioning of the classical moment sampling method and allows for more samples to be taken error bounds are derived for general polynomial sampling methods we also discuss various applications including thermodynamical properties of harmonic solids one striking result is the computation of the zero point vibrational energy of a face centered cubic crystal although our method does not provide upper and lower bounds constructing an ifs with 10 moments provides a comparable accuracy as the standard computation with 30 moments using pade approximations which are a special type of ifs

on the performance of fractal compression with clustering the paper investigates a technique to reduce the computational complexity of fractal image compression on gray scale images the technique uses a clustering process on image domain blocks with the clusters formed with the use of k d trees and the fast pairwise nearest neighbor algorithm of equitz 1984 results indicate the method is effective for smaller domain block sizes and generally shows improvement in terms of picture peak signal to noise ratio snr over the quadrant variance classification method

generalized dimensions entropies and liapunov exponents from the pressure function for strange sets for conformal mixing repellers such as julia sets and nonlinear one dimensional cantor sets we connect the pressure of a smooth transformation on the repeller with its generalized dimensions entropies and liapunov exponents computed with respect to a set of equilibrium gibbs measures this allows us to compute the pressure by means of simple numerical algorithms our results are then extended to axiom a attractors and to a nonhyperbolic invariant set of the line in this last case we show that a first order phase transition appears in the pressure

the box dimensions of inhomogeneous self conformal sets with overlaps abstract we investigate the box dimensions of inhomogeneous self conformal sets with overlaps firstly we significantly generalize the result of fraser concerning upper box dimension under some mild overlap conditions secondly we give corresponding formula of lower box dimension for condense set c with special construction these results extend the bowen s equation for inhomogeneous ifs

versal normal form at the lagrange equilibrium l4 a new normal form called versal for the linearized hamiltonian vector field of the planar restricted three body problem at the lagrange equilibrium point l4 depending smoothly on the mass ratio for all values close to the critical routh s ratio is described then a canonical transformation also depending smoothly on the mass ratio which brings the linear hamiltonian vector field into this versal normal form is explicitly calculated

image coding based on a fractal theory of iterated contractive image transformations the author proposes an independent and novel approach to image coding based on a fractal theory of iterated transformations the main characteristics of this approach are that i it relies on the assumption that image redundancy can be efficiently exploited through self transformability on a block wise basis and ii it approximates an original image by a fractal image the author refers to the approach as fractal block coding the coding decoding system is based on the construction for an original image to encode of a specific image transformation a fractal code which when iterated on any initial image produces a sequence of images that converges to a fractal approximation of the original it is shown how to design such a system for the coding of monochrome digital images at rates in the range of 0 5 1 0 b pixel the fractal block coder has performance comparable to state of the art vector quantizers

a mandelbrot set whose boundary is piecewise smooth it is proved that the mandelbrot set associated with the pair of maps w1 2 c c wi z sz 1 w2 z s z 1 with parameter s e c is connected and has piecewise smooth boundary introduction the discovery 1 of the mandelbrot set m for the iterated complex polynomial z2 s has generated considerable research activity 2 3 especially because of its relation to cascades of bifurcations and universal phenomena 4 the mandelbrot set m consists of those values of s e c such that the julia set j s for z2 s is connected barnsley and harrington 5 considered an analogous mandelbrot set d associated with the two affine maps t1 2 c c defined by t z sz 1 t2 z sz 1 for s e c and isi 1 there is a unique nonempty compact set a s which is invariant under t1 and t2 i e t1 a s ut2 a s a s 5 6 generically a s is a fractal d is defined to be the set of s e c isl 1 for which a s is disconnected the boundary of d contains self similar structures see figure 2 and appears to be a fractal it is not known whether d is connected however new pictures of this set presented here indicate that it is in this paper we study the mandelbrot set g associated with the two affine maps w1 c c defined by w1 z s2 11 w2 z s z 1 for s e c with isl 1 here s denotes the conjugate of s as in the previous case there is a unique invariant compact set a s which is generically a fractal despite the apparent similarity between the two pairs of maps g is easier to analyze than d we will show among other things that g is connected and remarkably has a piecewise smooth boundary pictures of the associated fractals as one travels around the boundary of g are given received by the editors november 23 1987 1980 mathematics subject classification 1985 revision primary 40a99 5 1m99 research by m f barnsley supported in part by darpa image compression project onr contract n0001 4 86 c 0446 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

vector potential normal form classification for completely integrable solenoidal nilpotent singularities abstract we introduce an sl 2 invariant family of polynomial vector fields with an irreducible nilpotent singularity in this paper we are concerned with characterization and normal form classification of these vector fields we show that the family is a lie subalgebra and each vector field from this family is volume preserving completely integrable and rotational all such vector fields share a common quadratic invariant we provide a poisson structure for the lie subalgebra from which the second invariant for each vector field can be readily derived we show that each vector field from this family can be uniquely characterized by two alternative representations which can be found in applications one uses a vector potential while the other uses two functionally independent clebsch potentials our normal form results are designed to preserve these structures and representations

orthogonal polynomials of equilibrium measures supported on cantor sets the equilibrium measure of a compact set is a fundamental object in logarithmic potential theory we compute numerically this measure and its orthogonal polynomials when the compact set is a cantor set defined by an iterated function system we first construct sequences of discrete measures via the solution of large systems of non linear equations that converge weakly to the equilibrium measure successively we compute their jacobi matrices via standard procedures enhanced for the scope numerical estimates of the convergence rate to the limit jacobi matrix are provided which show stability and efficiency of the whole procedure as a companion result we also compute jacobi matrices in two other cases equilibrium measures on finite sets of intervals and balanced measures of iterated function systems these algorithms can reach large polynomial orders therefore we study the asymptotic behavior of the orthogonal polynomials and by a natural extension of the concept of regular root asymptotics we derive an efficient scheme for the computation of complex green s functions and of related conformal mappings

nonequilibrium glauber type dynamics in continuum we construct the nonequilibrium glauber dynamics as a markov process in configuration space for an infinite particle system in continuum with a general class of initial distributions this class we define in terms of correlation functions bounds and it is preserved during the markov evolution we constructed

self affine fractal functions and wavelet series abstract we consider functions represented by series g g c g g 1 x of wavelet type where g is a group generated by affine functions l 1 l n and is piecewise affine by means of those functions we characterize the class of self affine fractal functions previously studied by barnsley et al we compute their global and local holder exponents and investigate points of non differentiability wavelet representations for various continuous nowhere differentiable and singular functions are presented another application is the construction of functions with prescribed local holder exponents at each point

a neural network to compute the hutchinson metric in fractal image processing the hutchinson metric is a natural measure of the discrepancy between two images for use in fractal image processing a neural network is described which can quickly calculate this metric by combining this with the architecture previously described by the author for implementing the markov operator of an iterated function system ifs on a neural network a fast method is obtained for determining the distance between a target image and the invariant measure of a trial ifs this has obvious applications to barnsley s fractal image compression scheme

a fast encoding algorithm for fractal image compression using the dct inner product in this paper a fast encoding algorithm is developed for fractal image compression at each search entry in the domain pool the mean square error mse calculations of the given range block and the eight dihedral symmetries of the domain block are obtained simultaneously in the frequency domain in which the redundant computations are all eliminated in the new encoding algorithm it is shown in software simulation that the encoding time is about six times faster than that of the baseline method with almost the same psnr for the retrieved image the fast algorithm is performed to deal with the eight dihedral symmetries at each search entry therefore it can be applied to various enhanced algorithms which are equipped with quadtree classification and other mechanisms

wave propagation in almost periodic structures abstract the unusual phenomena occurring in the wave propagation in almost periodic structures both classical and quantum are studied in this paper focusing our attention on structures with spectral measures in the family of disconnected iterated function systems we describe and characterize the concept of quantum intermittency this theory shows that the nontrivial renormalization properties of the set of orthogonal polynomials associated with these systems are the origin of such intermittency and leads to a new determination of the exponents of the asymptotic growth of the moments of the position operator

moments of balanced measures on julia sets by a theorem of s demko there exists a balanced measure on the julia set of an arbitrary nonlinear rational transformation on the riemann sphere it is proved here that if the transformation admits an attractive or indifferent cycle then there is a point with respect to which all the moments of a balanced measure exist moreover these moments can be calculated exactly an explicit balanced measure is exhibited in an example where the julia set is the whole sphere and for which the moments with respect to any point do not all exist 1 introduction the julia set for a nonlinear rational transformation on the riemann sphere is the closure of the set of all repulsive cycles of the transformation br fa j the location or distribution of the set is of interest in the theory of iterated maps and it may be characterized by an associated invariant measure in this paper we extend previous results by showing that when there is an attractive or indifferent cycle certain moments of such a measure exist and can be calculated exactly from the rational transformation this means for example that when the julia set is contained strictly in the real line all of the orthogonal polynomials on the set can be calculated as can the entries of the corresponding infinite dimensional jacobi matrices the latter are of the type which have recently been shown to furnish model schrodinger equations for crystals with almost periodic potentials bbm this leads us to the following speculation which provides one reason for our interest in the present topic we note first the frequent occurrence in the physical universe of objects steady state systems which are fractals from some point of

thermodynamics of fractal signals based on wavelet analysis application to fully developed turbulence data and dna sequences we use the continuous wavelet transform to generalize the multifractal formalism to fractal functions we report the results of recent applications of the so called wavelet transform modulus maxima wtmm method to fully developed turbulence data and dna sequences we conclude by briey describing some works currently under progress which are likely to be the guidelines for future research c 1998 elsevier science b v all rights reserved

transformation to versal deformations of matrices abstract in the paper versal deformations of matrices are considered the versal deformation is a matrix family inducing an arbitrary multi parameter deformation of a given matrix by an appropriate smooth change of parameters and basis given a deformation of a matrix it is suggested to find transformation functions the change of parameters and the change of basis dependent on parameters in the form of taylor series the general method of construction of recurrent procedures for calculation of coefficients in the taylor expansions is developed and used for the cases of real and complex matrices elements of classical lie and jordan algebras and infinitesimally reversible matrices several examples are given and studied in detail applications of the suggested approach to problems of stability singularity and perturbation theories are discussed

normal form theory and spectral sequences abstract the concept of unique normal form is formulated in terms of a spectral sequence as an illustration of this technique some results of baider and churchill concerning the normal form of the anharmonic oscillator are reproduced the aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory

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

a generalized fractal transform for measure valued images abstract fractal image coding generally seeks to express an image as a union of spatially contracted and greyscale modified copies of subsets of itself generally images are represented as functions u x and the fractal coding method is conducted in the framework of 2 or in this paper we formulate a method of fractal image coding on measure valued images at each point x x is a probability measure over the range of allowed greyscale values we construct a complete metric space y d y of measure valued images x m r g where x is the base or pixel space and m r g is the set of probability measures supported on the greyscale range r g a generalized fractal transform m is formulated over the metric space y d y under suitable conditions m y y is contractive implying the existence of a unique fixed point measure valued function m

generalized dimensions large deviations and the distribution of rare events abstract generalized dimensions of multifractal measures are usually seen as static objects related to the scaling properties of suitable partition functions or moments of measures of cells when these measures are invariant for the flow of a chaotic dynamical system generalized dimensions take on a dynamical meaning as they provide the rate function for the large deviations of the first hitting time which is the average time required to connect any two different regions in phase space we prove this result rigorously under a set of stringent assumptions as a consequence the statistics of hitting times provides new algorithms for the computation of the spectrum of generalized dimensions numerical examples presented along with the theory suggest that the validity of this technique reaches far beyond the range covered by the theorem we state our result within the framework of extreme value theory this approach reveals that hitting times are also linked to dynamical indicators such as stability of the motion and local dimensions of the invariant measure this suggests that one can use local dynamical indicators from finite time series to gather information on the multifractal spectrum of generalized dimension we show an application of this technique to experimental data from climate dynamics

an analogue of cobham s theorem for graph directed iterated function systems abstract feng and wang showed that two homogeneous iterated function systems in r with multiplicatively independent contraction ratios necessarily have different attractors in this paper we extend this result to graph directed iterated function systems in r n with contraction ratios that are of the form 1 for integers by using a result of boigelot et al this allows us to give a proof of a conjecture of adamczewski and bell in doing so we link the graph directed iterated function systems to buchi automata in particular this link extends to real numbers we introduce a logical formalism that permits to characterize sets of r n whose representations in base are recognized by some buchi automata this result depends on the algebraic properties of the base being a pisot or a parry number the main motivation of this work is to draw a general picture representing the different frameworks where an analogue of cobham s theorem is known

inverse and approximation problem for two dimensional fractal sets the geometry of fractals is rich enough that they have extensively been used to model natural phenomena and images iterated function systems ifs theory provides a convenient way to describe and classify deterministic fractals in the form of a recursive definition as a result it is conceivable to develop image representation schemes based on the ifs parameters that correspond to a given fractal image in this paper we consider two distinct problems an inverse problem and an approximation problem the inverse problem involves finding the ifs parameters of a signal that is exactly generated via an ifs we make use of the wavelet transform and of the image moments to solve the inverse problem the approximation problem involves finding a fractal ifs generated image whose moments match either exactly or in a mean squared error sense a range of moments of the original image the approximating measures are generated by an ifs model of a special form and provide a general basis for the approximation of arbitrary images experimental results verifying our approach will be presented

study on huber fractal image compression in this paper a new similarity measure for fractal image compression fic is introduced in the proposed huber fractal image compression hfic the linear huber regression technique from robust statistics is embedded into the encoding procedure of the fractal image compression when the original image is corrupted by noises we argue that the fractal image compression scheme should be insensitive to those noises presented in the corrupted image this leads to a new concept of robust fractal image compression the proposed hfic is one of our attempts toward the design of robust fractal image compression the main disadvantage of hfic is the high computational cost to overcome this drawback particle swarm optimization pso technique is utilized to reduce the searching time simulation results show that the proposed hfic is robust against outliers in the image also the pso method can effectively reduce the encoding time while retaining the quality of the retrieved image

efficiency speed up strategies for evolutionary computation fundamentals and fast gas through identifying the main causes of low efficiency of the currently known evolutionary algorithms a set of six efficiency speed up strategies are suggested analyzed and partially explored including those of the splicing decomposable representation scheme the exclusion based selection operators the few generation ended ec search the low resolution computation with reinitialization and the coevolution like decomposition incorporation of the strategies with any known evolutionary algorithm leads to an accelerated version of the algorithm on the basis of problem space discretization the proposed strategies accelerate evolutionary computation via a best so far solution guided exclusion based space shrinking scheme with this scheme an arbitrarily high precision resolution solution of a high dimensional problem can be obtained by means of a successive low resolution search in low dimensional search spaces as a case study the developed strategies have been endowed with genetic algorithms gas yielding an accelerated genetic evolutionary algorithm the fast gas the fast gas are experimentally tested with a test suit containing 10 complex multimodal function optimization problems and a difficult real life problem the moment matching problem for inverse to the fractal encoding of the spleenwort fern the performance of the fast ga is compared against the standard ga sga and the forking ga fga that is one of the most recent and fairly established variants of gas the experiments all demonstrate that the fast gas consistently and significantly outperform the sga and fga in efficiency and solution quality in the test cases besides the speed up of efficiency other visible features of the fast gas include i no premature convergence occurs ii better convergence capability to global optimum and iii variable high precision solutions attainable all these support the validity and usefulness of the developed strategies

v variable fractals fractals with partial self similarity we establish properties of a new type of fractal which has partial self similarity at all scales for any collection of iterated functions systems with an associated probability distribution and any positive integer v there is a corresponding class of v variable fractal sets or measures these v variable fractals can also be obtained from the points on the attractor of a single deterministic iterated function system existence uniqueness and approximation results are established under average contractive assumptions we also obtain extensions of some basic results concerning iterated function systems

the randomly driven ising ferromagnet i general formalism and mean field theory we consider the behaviour of an ising ferromagnet obeying the glauber dynamics under the influence of a fast switching random external field after introducing a general formalism for describing such systems we consider here the mean field theory a novel type of first order phase transition related to spontaneous symmetry breaking and dynamic freezing is found the nonequilibrium stationary state has a complex structure which changes as a function of parameters from a singular continuous distribution with euclidean or fractal support to an absolutely continuous one these transitions are reflected in both finite size effects and sample to sample fluctuations

the superstability of pair potentials of positive type we prove that a pair potential which is continuous l1 and of positive type satisfies a condition of the superstability kind with best possible constants the applications to statistical thermodynamics are mentioned

power domains and iterated function systems we introduce the notion of weakly hyperbolic iterated function system ifs on a compact metric space which generalises that of hyperbolic ifs based on a domain theoretic model which uses the plotkin power domain and the probabilistic power domain respectively we prove the existence and uniqueness of the attractor of a weakly hyperbolic ifs and the invariant measure of a weakly hyperbolic ifs with probabilities extending the classic results of hutchinson for hyperbolic ifss in this more general setting we also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure extending the corresponding algorithms for hyperbolic ifss we then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent ifs and obtain an algorithm to generate the invariant distribution on the digitised screen the generalised riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution for hyperbolic recurrent ifss and lipschitz maps one can estimate the integral up to any threshold of accuracy

contractive multifunctions fixed point inclusions and iterated multifunction systems abstract we study the properties of multifunction operators that are contractive in the covitz nadler sense in this situation such operators t possess fixed points satisfying the relation x t x we introduce an iterative method involving projections that guarantees convergence from any starting point x 0 x to a point x x t the set of all fixed points of a multifunction operator t we also prove a continuity result for fixed point sets x t as well as a generalized collage theorem for contractive multifunctions these results can then be used to solve inverse problems involving contractive multifunctions two applications of contractive multifunctions are introduced i integral inclusions and ii iterated multifunction systems

a new approximation procedure for fractals this paper is based upon hutchinson s theory of generating fractals as fixed points of a finite set of contractions when considering this finite set of contractions as a contractive set valued map we approximate the fractal using some preselected parameters and we obtain formulae describing the distance between the exact fractal and the approximate fractal in terms of the preselected parameters some examples and also computation programs are given showing how our procedure works

recurrent random walks in nonnegative matrices attractors of certain iterated function systems in this paper the structure of the set of recurrent points for random walks in finite dimensional nonnegative matrices is determined the structural results are then used in understanding attractors of certain not necessarily contractive iterated function systems

the rate distortion dimension of sets and measures data compression of independent samples drawn from a fractal set is considered the asymptotic ratio of rate to magnitude log distortion characterizes the effective dimension occupied by the underlying distribution this quantity is shown to be identical to renyi s 1959 information dimension for self similar fractal sets this dimension is distribution dependent in sharp contrast with the behavior of absolutely continuous measures the rate distortion dimension of a set is defined as the maximal rate distortion dimension for distributions supported on this set kolmogorov s metric dimension is an upper bound on the rate distortion dimension while the hausdorff dimension is a lower bound examples of sets for which the rate distortion dimension differs from these bounds are provided