a variational principle for finite planar deformation of straight slender elastic beams abstract the objective of this paper is to present a finite element formulation for finite deformation analysis of a slender straight elastic in plane beam the formulation is based on a modified hu washizu variational principle in which exact nonlinear kinematic equations are taken into account only the rotation in the interior of an element needs to be approximated thus the ambiguities concerning the order of polynomial approximations for different field variables e g longitudinal and lateral displacements are avoided the euler lagrange equations of this principle are among others exact kinematic and equilibrium conditions for the beam the solution capabilities are illustrated with numerical examples two different finite elements are examined with 3rd order interpolation polynomials for the rotation and with 5th order polynomials excellent convergence and accuracy of both elements is demonstrated the results indicate that the accuracy of the present elements is not notably influenced by the length of the element and the order of numerical integration if it is greater than a minimal value only one load step was sufficient in calculations to achieve the results

optimization of mechanical systems on strategy of non linear first order approximation the paper presents an extended approach to non linear first order approximation of non linear programming problems and it explains how to transform an approximate problem into a strictly convex one the essence of the proposed approximation technique is to rewrite each given function hj as a composite gjo j the function j has to be chosen the paper explains how to do this while gj is linear approximated with gj the approximation of hj is then obtained as gjo j this approach enables one to obtain approximate functions with variable conservativeness which implies an adjustable approximate problem a solution procedure which replaces the original problem with a sequence of approximate problems can therefore adjust each succeeding approximate problem to improve the convergence properties the theory is illustrated with a three parameters controlled approximation this technique represents together with an optimality criteria based solution procedure a powerful and economic tool for solving nonlinear programming problems the three parameters which influence to a great extent the conservativeness of the approximate functions are under full control of the optimizer they are varied automatically during the process of optimization to speed up the convergence or to prevent oscillations the benefits gained from the proposed approach are demonstrated on several numerical examples involving structures and a dynamic multibody system

random recursive construction of self similar fractal measures the noncompact case the self similarity of sets measures is often defined in a constructive way in the present paper it will be shown that the random recursive construction model of falconer graf and mauldin williams for statistically self similar sets may be generalized to the noncompact case we define a sequence of random finite measures which converges almost surely to a self similar random limit measure under certain conditions on the generating lipschitz maps we determine the carrying dimension of the limit measure

an efficient approximation technique for frequency constraints in frame optimization an efficient approximation technique is presented for dealing with frequency constraints in frame optimization the method is based on a linear approximation of the eigenvalues with respect to the member section properties using this an approximate optimization problem is solved treating the physical dimensions of the beams as design variables when the eigenvalues are required the section properties are first calculated then using the approximation the eigenvalues are estimated and returned to the optimizer when the approximate optimum is found a detailed analysis is performed and the process is repeated examples are presented to demonstrate the method the method is simple yet treats the physical dimensions of the structure as design variables and allows for large move limits during the approximate optimization

inversion of an integral transform related to a general form of heat equation inversion algorithms have been derived by d v widder for the weierstrass transform and by d t haimo for the dual weierstrass laguerre transform for the genera1 form of heat equation l u u l a self adjoint operator an integral transform is introduced with kernel related to the fundamental solution of the equation an inversion formula for the transform is derived which includes the previous resuks as special cases cf 1988 academic press inc

statistically self similar fractals letx be a complete separable bounded metric space and a borel probability measure on the space con x n of alln tuples of contractions ofx with the topology of pointwise convergence then there exists a unique self similar probability measurep on the spacek x of all non empty compact subsets ofx here a measurep onk x is called self similar if for every borel setb k x p b int p n left k o k n 1 left bigcup limits i 0 n 1 s i k i in b right right d mu s o s n 1 if for a e s 0 sn 1 eachs i has an inverse which satisfies a lipschitz condition then there is an 0 such that forp a e k k x the hausdorff dimensionh dim k is equal to ifx d is compact and has non empty interior and if a e s 0 sn 1 consists of similarities which satisfy a certain disjointness condition w r t x then is determined by the equation int sum limits i 0 n 1 lip s i alpha d mu s o s n 1 1 where lip s i denotes the smallest lipschitz constant fors i under fairly general assumptions the dimensional hausdorff measure ofp a e k k x equals o if andx are chosen in a rather special way thenp a e k k x is the graph of a homeomorphism of 0 1 or a curve or the graph of a continuous function

on the dimension of deterministic and random cantor like sets symbolic dynamics and the eckmann ruelle conjecture in this paper we unify and extend many of the known results on the dimension of deterministic and random cantor like sets in n and apply these results to study some problems in dynamical systems in particular we verify the eckmann ruelle conjecture for equilibrium measures for holder continuous conformal expanding maps and conformal axiom a topologically hyperbolic homeomorphims we also construct a holder continuous axiom a homeomorphism of positive topological entropy for which the unique measure of maximal entropy is ergodic and has different upper and lower pointwise dimensions almost everywhere this example shows that the non conformal holder continuous version of the eckmann ruelle conjecture is false

the exact hausdorff dimension of a branching set we obtain a critical function for which the hausdorff measure of a branching set generated by a simple galton watson process is positive and finite

dimensions and singular traces for spectral triples with applications to fractals given a spectral triple a h d the functionals on a of the form a a d are studied where is a singular trace and is a generalised limit when is the dixmier trace the unique exponent d giving rise possibly to a non trivial functional is called hausdorff dimension and the corresponding functional the d dimensional hausdorff functional it is shown that the hausdorff dimension d coincides with the abscissa of convergence of the zeta function of d 1 and that the set of s for which there exists a singular trace giving rise to a non trivial functional is an interval containing d moreover the endpoints of such traceability interval have a dimensional interpretation the functionals corresponding to points in the traceability interval are called hausdorff besicovitch functionals these definitions are tested on fractals in r by computing the mentioned quantities and showing in many cases their correspondence with classical objects in particular for self similar fractals the traceability interval consists only of the hausdorff dimension and the corresponding hausdorff besicovitch functional gives rise to the hausdorff measure more generally for any limit fractal the described functionals do not depend on the generalized limit

inversion of integral transforms associated with a class of perturbed heat equations abstract let lx be the sturm liouville differential operator l x d 2 dx 2 q x x 0 we assume that lx has either a purely discrete spectrum that is bounded from below by zero or a continuous spectrum that fills up the interval 0 with possibly a finite number of negative eigenvalues the w transform of y l2 0 is defined by x 0 y g x y 1 dy where g x y t is a function associated with the fundamental solution of the perturbed equation l x u x t u x t t the main purpose of this paper is to derive an inversion formula for the w transform this inversion formula generalizes the known inversion formulae for the weierstrass weierstrass hankel convolution and weierstrass laguerre transforms the results of this paper are easily extended to the case where lx is considered over the entire line

self similar random measures a set is called self similar if it is decomposable into parts which are similar to the whole this notion was generalized to random sets in the present paper an alternative axiomatic approach is given which makes precise the following idea using palm distribution theory a random set is statistically self similar if it is statistically scale invariant with respect to any center chosen at random from that set for these sets hausdorff dimension coincides with an intrinsic self similarity index

the dimension of projections of fractal percolations fractal percolation or mandelbrot percolation is one of the most well studied families of random fractals in this paper we study some of the geometric measure theoretical properties dimension of projections and structure of slices of these random sets although random the geometry of those sets is quite regular our results imply that denoting by e subset mathbb r 2 a typical realization of the fractal percolation on the plane if dim rm h e 1 then for all lines l the orthogonal projection el of e to l has the same hausdorff dimension as e if dim rm h e 1 then for any smooth real valued function f which is strictly increasing in both coordinates the image f e contains an interval the second statement is quite interesting considering the fact that e is almost surely a cantor set a random dust for a large part of the parameter domain see chayes et al probab theory relat fields 77 3 307 324 1988 finally we solve a related problem about the existence of an interval in the algebraic sum of d 2 one dimensional fractal percolations

central limit theorems for the spectra of classes of random fractals we discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal the continuum random tree in the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit we will show examples from a class of random fractals generated from dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem the brownian continuum random tree can also be viewed as a random fractal generated by a dirichlet distribution the first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this though the positivity of the variance arising in the central limit theorem is left open in both cases these fractals can be described through a general crump mode jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known

exact packing dimension in random recursive constructions we explore the exact packing dimension of certain random recursive constructions in case of polynomial decay at 0 of the distribution function of random variable x associated with the construction we prove that it does not exist and in case of exponential decay it is t log logt where is the fractal dimension of the limit set and 1 is the rate of exponential decay

connectivity properties of mandelbrot s percolation process in 1974 mandelbrot introduced a process in 0 1 2 which he called canonical curdling and later used in this book s on fractals to generate self similar random sets with hausdorff dimension d 0 2 in this paper we will study the connectivity or percolation properties of these sets proving all of the claims he made in sect 23 of the fractal geometry of nature and a new one that he did not anticipate there is a probability pc 0 1 so that if p pc then the set is duslike i e the largest connected component is a point whereas if p pc notice the opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles more succinctly put the system has a first order phase transition

using iterated function systems to model discrete sequences two iterated function system ifs models are explored for the representation of single valued discrete time sequences the self affine fractal model and the piecewise self affine fractal model algorithms are presented one of which is suitable for a multiprocessor implementation for identification of the parameters of each model applications of these models to a variety of data types are given where signal to noise ratios are presented quantization effects of the model parameters are investigated and compression ratios are computed

random fractal strings their zeta functions complex dimensions and spectral asymptotics in this paper a string is a sequence of positive non increasing real numbers which sums to one for our purposes a fractal string is a string formed from the lengths of removed sub intervals created by a recursive decomposition of the unit interval by using the so called complex dimensions of the string the poles of an associated zeta function it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string we consider random versions of fractal strings we show that by using a random recursive self similar construction it is possible to obtain similar results to those for deterministic self similar strings in the case of strings generated by the excursions of stable subordinators we show that the complex dimensions can only lie on the real line the results allow us to discuss the geometric and spectral asymptotics of one dimensional domains with random fractal boundary

valuations of languages with applications to fractal geometry abstract valuations morphisms from to 0 1 are a simple generalization of bernoulli morphisms distributions measures as introduced by berstel and perrin 1988 berstel and reutenauer 1988 blanchard and hansel 1984 eilenberg 1974 and hansel and perrin 1983 1989 this paper shows that valuations are useful not only within the theory of codes but also when dealing with ambiguity especially in context free grammars or for defining outer measures on the space of words which are of some importance to the theory of fractals these connections yield new formulae to determine the hausdorff dimension of fractal sets especially in euclidean spaces defined via formal languages the class of fractals describable with context free languages strictly includes that of mrfs fractals introduced by culik and dube 1990 and mauldin and williams 1988 some of the results of this paper also appear as part of the author s ph d thesis and other works fernau 1992a b 1993

new concepts for finite element mass matrix formulations improved finite element formulations are presented for dynamic analysis the improvements are based on modifications of the finite element mass matrix and are of high importance since no additional computational effort is required compared with the necessary effort for the standard consistent mass matrix formulation formulations are presented for the one dimensional bar two dimensional membrane and the pure bending beam element free vibration analysis and forced vibration results are presented employing new mass matrices in general the frequency errors and stress errors are at least three times smaller than the corresponding consistent or lumped mass formulation errors accurate model data can be achieved for all the modes of relatively coarse meshes

recursive partition structures a class of random discrete distributions p is introduced by means of a recursive splitting of unity assuming supercritical branching we show that for partitions induced by sampling from such p a power growth of the number of blocks is typical some known and some new partition structures appear when p is induced by a dirichlet splitting

brownian motion on a random recursive sierpinski gasket we introduce a random recursive fractal based on the sierpinski gasket and construct a diffusion upon the fractal via a dirichlet form this form and its symmetrizing measure are determined by the electrical resistance of the fractal the effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion the main result is to obtain uniform upper and lower bounds for the transition density of the brownian motion on the fractal in terms of this metric the bounds are not tight as there are logarithmic corrections due to the randomness in the environment and the behavior of the shortest paths in the effective resistance metric is not well understood the results are deduced from the study of a suitable general branching process

the nature of the atomic surfaces of quasiperiodic self similar structures quasiperiodic self similar chains generated by substitutions i e deterministic concatenation rules and their diffraction spectra are analysed in a systematic fashion from the viewpoint of the superspace formalism a substitution acting on n objects generates quasiperiodic chains if and only if the associated substitution matrix fulfils two arithmetic conditions pisot property and unit determinant the structures thus obtained can be alternatively built as sections of periodic patterns in an n dimensional superspace which are regular repetitions of an atomic surface the authors derive a general algorithm to construct this atomic surface it is a compact set of the n 1 dimensional internal space which is a unit cell for a lattice of translations the atomic surface is nevertheless not necessarily connected and its boundary is generically an anisotropic self similar fractal the dimension db of this boundary is shown to govern the anomalously slow fall off of the intensities of bragg diffractions and therefore to influence physical properties

log similarity for turbulent flows abstract we critically examine a recently proposed representation for turbulent energy spectra called here log similarity which is predicted both by the multifractal model and the variational approach of the small scale intermittency within this representation spectra are homothetic in the log log plot the homothety factor depending on the taylor scale based reynolds number r experimental results on energy spectra and velocity structure functions of order 2 3 and 6 measured in different flows at several r show that this factor is r a measurable exponent which characterises intermittency in the inertial range experimentally displays the existence of a transition in the neighbourhood of r 400 and for r tending to infinity behaves as 1 in r r

the dual weierstrass laguerre transform an inversion algorithm is derived for the dual weierstrass laguerre transform 1o g x y 1 9 v yae a l dy where the function g x x y t is associated with the source solution of the laguerre differential heat equation xuv y x t a 1 x u x t u x t correspondingly sufficient conditions are established for a function to be represented by a weierstrass laguerre stieltjes transform 1o ga x y 1 d 8 y of a nondecreasing function b

galton watson iterated function systems iterated function systems ifs are interesting parametric models for generating fractal sets and functions the general idea is to compress deform and translate a given set or function with a collection of operators and to iterate the procedure under weak assumptions ifs possess a unique fixed point which is in general fractal ifs were introduced in a deterministic context then were generalized to the random setting on abstract spaces in the early 1980 s their adaptation to random signals was carried out by hutchinson and r schendorff 9 by considering random operators this study extends their model with not only random operators but also a random underlying construction tree we show that the corresponding ifs converges under certain hypothesis to a unique fractal fixed point properties of the fixed point are also described

superbranching processes and projections of random cantor sets we study sequences x 0 x 1 of random variables taking values in the positive integers which grow faster than branching processes in the sense that x m n geqq sum limits i 1 x m x n m i for m n 0 where the x n m i are distributed as x n and have certain properties of independence we prove that under appropriate conditions x 1 almost surely and in l 1 where sup e x n 1 n our principal application of this result is to study the lebesgue measure and hausdorff dimension of certain projections of sets in a class of random cantor sets being those obtained by repeated random subdivisions of the m adic subcubes of 0 1 d we establish a necessary and sufficient condition for the lebesgue measure of a projection of such a random set to be non zero and determine the box dimension of this projection

pseudononstationarity in the scaling exponents of finite interval time series the accurate estimation of scaling exponents is central in the observational study of scale invariant phenomena natural systems unavoidably provide observations over restricted intervals consequently a stationary stochastic process time series can yield anomalous time variation in the scaling exponents suggestive of nonstationarity the variance in the estimates of scaling exponents computed from an interval of n observations is known for finite variance processes to vary as ensuremath sim 1 n as n ensuremath rightarrow ensuremath infty for certain statistical estimators however the convergence to this behavior will depend on the details of the process and may be slow we study the variation in the scaling of second order moments of the time series increments with n for a variety of synthetic and real world time series and we find that in particular for heavy tailed processes for realizable n one is far from this ensuremath sim 1 n limiting behavior we propose a semiempirical estimate for the minimum n needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some real world time series

on integral transforms whose kernels are solutions of singular sturm liouville problems on etudie des transformees integrales du type f x 0 f s x s d ou x s est la solution du probleme de sturm liouville y s 2 q x y 0 0 x avec y 0 cos y 0 sin 0 y x est bornee a l infini et d est la mesure spectrale

hausdorff dimension of wild fractals we show that for every s n 2 n there exists a homogeneously embedded wild cantor set c s in r n n 3 of local hausdorff dimension s also it is shown that for every s n 2 n and for any integer k n such that 1 k s there exist everywhere wild k spheres and k cells in r n n 3 of local hausdorff dimension s

shape optimal design of elastic space frames with non linear response this paper describes an approach to shape optimal design of elastic planar frames with non linear response the foundation of the proposed approach forms an appropriate strategy of shape representation of the structure based on the design element technique a frame structure is treated as to be assembled from several frame design elements which in turn may consist of several appropriately joined beam finite elements the shape of each frame design element is defined by convenient functions involving bezier blending polynomials the original formulation of the beam finite element proposed by saje is modified in order to fit nicely into the context of the frame design element technique the formulation of the shape optimal design problem in a form of a problem of non linear mathematical programming and its solution by employing gradient based methods of mathematical programming are discussed briefly the theory is illustrated with two numerical examples

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

design sensitivity analysis and optimization of nonlinear structuralresponse using incremental procedure this paper describes procedures for design sensitivity analysis of nonlinear structural response using the incremental solution scheme the total and updated lagrangian approaches are discussed sensitivity analyses for the stress strain and displacement constraints are developed in the optimization process the buckling load constraint is also imposed numerical aspects of design sensitivity analysis are discussed three numerical examples are solved to illustrate the design sensitivity analysis and optimization process the numerical procedures are stable and work well they can be applied to more complex structures to optimize them

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

recursive self similarity for random trees random triangulations and brownian excursion recursive self similarity for a random object is the property of being decomposable into independent rescaled copies of the original object certain random combinatorial objects trees and triangulations possess approximate versions of recursive self similarity and then their continuous limits possess exact recursive self similarity in particular since the limit continuum random tree can be identified with brownian excursion we get a nonobvious recursive self similarity property for brownian excursion

dynamic optimization of framed structures with variable layout the paper presents a solution to the problem of dynamic optimization of framed structures for which the design variables are not only the sizes of the members but also the positions of the joints the optimal design obtained is the least weight design which satisfies a specific frequency requirement plus upper and lower bounds on the design variables the design algorithm is an iterative solution of the kuhn tucker optimality criterion the procedure is to modify an initial design to first obtain the desired fundamental frequency and then while this frequency is held constant to minimize the weight the most important features of the algorithm are a a small number of design iterations is needed to reach an optimal or near optimal design b structural elements with a wide variety of size stiffness may be used the design procedure is illustrated with some numerical examples the results show that the method is rapidly convergent and the optimized configurations are determined in a small number of redesign cycles the procedure has been completely automated in a computer program on the hp 45 desk top computer

shaping optimal design of elastic planar frames with frequency constraints an approach to shaping optimal design of planar frames subject to eigenfrequency constraints displacement constraints and stress constraints simultaneously is described the eigenfrequency constraints may be imposed at the deformed due to the static load cone guration of the structure for the shape representation of the structure a design element technique is used a frame structure is treated as assembled from several design elements each of them being dee ned as a bezier curve the design variables may ine uence the position and the shape of each design element plus the cross sectional properties of beam elements the optimal design problem is dee ned in a general form and its solution by employing gradient based methods of mathematical programming is discussed briee y the structural response analysis is based on a nonlinear model that can handle virtually arbitrary large structural displacements two numerical examples illustrate the theory nomenclature ai area of the cross section asi shear area of the cross section b vector of design variables ci centroid curve e elastic modulus of the material f vector of internal forces fi eigenfrequencies g shear modulus of the material hi constrained quantitites ii moment of inertia of the cross section k stiffness matrix m mass matrix pi lagrange interpolation polynomial p external distributed force per unit length qi position vector of a control point r vector of external forces ri position vector u vector of structural displacements v vector of structural response variables w external distributed force per unit length shear strain extensional strain i eigenvalues angle of inclination at the undeformed cone guration i eigenvector a angle of rotation

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

lacunarity of self similar and stochastically self similar sets let k be a self similar set in rdd of hausdorff dimension d and denote by ik c l the d dimensional lebesgue measure of its 6 neighborhood we study the limiting behavior of the quantity e d d ik e i as 6 e 0 it turns out that this quantity does not have a limit in many interesting cases including the usual ternary cantor set and the sierpinski carpet we also study the above asymptotics for stochastically self similar sets the latter results then apply to zero sets of stable bridges which are stochastically self similar in the sense of the present paper and then more generally to level sets of stable processes specifically it follows that if kt is the zero set of a realvalued stable process of index a c 1 2 run up to time t then e siit e i converges to a constant multiple of the local time at 0 simultaneously for all t 0 on a set of probability one the asymptotics for deterministic sets are obtained via the renewal theorem the renewal theorem also yields asymptotics for the mean 1e ik c in the random case while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks

branching random walk with exponentially decreasing steps and stochastically self similar measures we consider a branching random walk on r whose step size decreases by a fixed factor 0 1 2 the limit measure is almost surely a s absolutely continuous with respect to the lebesgue measure but for pisot 1 it is a s singular 2 for all a 5 1 2 the support of the measure is a s the closure of its interior 3 for pisot 1 the support of the measure is fractured it is a s disconnected and the components of the complement are not isolated on both sides

wavelet based estimation of 1 f type signal parameters confidence intervals using the bootstrap we propose to construct confidence intervals of parameters of 1 f type signals using a nonparametric wavelet based bootstrap method bootstrap based confidence intervals of maximum likelihood parameter estimates are compared to the confidence intervals derived from the cramer rao lower bound crlb for moderately large data sample sizes the bootstrap approach achieves the nominal coverage and may perform better than the crlb based parametric approach

data driven detrending of nonstationary fractal time series with echo state networks abstract in this paper we propose a novel data driven approach for removing trends detrending from nonstationary fractal and multifractal time series we consider real valued time series relative to measurements of an underlying dynamical system that evolves through time we assume that such a dynamical process is predictable to a certain degree by means of a class of recurrent networks called echo state network esn which are capable to model a generic dynamical process in order to isolate the superimposed multi fractal component of interest we define a data driven filter by leveraging on the esn prediction capability to identify the trend component of a given input time series specifically the estimated trend is removed from the original time series and the residual signal is analyzed with the multifractal detrended fluctuation analysis procedure to verify the correctness of the detrending procedure in order to demonstrate the effectiveness of the proposed technique we consider several synthetic time series consisting of different types of trends and fractal noise components with known characteristics we also process a real world dataset the sunspot time series which is well known for its multifractal features and has recently gained attention in the complex systems field results demonstrate the validity and generality of the proposed detrending method based on esns

thermodynamics for the zero level set of the brownian bridge the random set of instants where the brownian bridge vanishes is constructed in terms of a random branching process the hausdorff measure supported by this set is shown to be equivalent to the partition function of a special class of disordered systems this similarity is used to show rigorously the existence of a phase transition for this particular class of disordered systems moreover it is shown that at high temperature the specific free energy has the strong self averaging property and that at low temperature it has no self averaging property the unicity at high temperature and the existence of many limits at low temperature are established almost surely in the disorder

how much information does dependence between wavelet coefficients contain abstractthis article is motivated by several articles that propose statistical inference where the independence of wavelet coefficients for both short as well as long range dependent time series is assumed we focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic to this end we derive asymptotic distributional properties of the sample variance for a time series that is synthesized ignoring some or all dependence between wavelet coefficients we show that the second order properties differ from the those of the true time series whose wavelet coefficients have the same marginal distribution except in the independent gaussian case this holds true even if the dependency is correct within each level and only the dependence between levels is ignored in the case of sample autocovariances and sample autocorrelations at lag one we indicate that first order properties are erroneous in a second step we investigate several nonparametric b

self similar anisotropic texture analysis the hyperbolic wavelet transform contribution textures in images can often be well modeled using self similar processes while they may simultaneously display anisotropy the present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of gaussian anisotropic selfsimilar processes it will be first shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2d discrete wavelet transform with the hyperbolic wavelet transform which permits the use of different dilation factors along the horizontal and vertical axes defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized this discrepancy defines a rotation angle second we show that this rotation angle can be jointly estimated third a nonparametric bootstrap based procedure is described which provides confidence intervals in addition to the estimates themselves and enables us to construct an isotropy test procedure which can be applied to a single texture image fourth the robustness and versatility of the proposed analysis are illustrated by being applied to a large variety of different isotropic and anisotropic self similar fields as an illustration we show that a true anisotropy built in self similarity can be disentangled from an isotropic self similarity to which an anisotropic trend has been superimposed

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

self avoiding process on the sierpinski gasket we construct a self avoiding process taking values in the finite sierpinski gasket and study its properties we then study continuum limit processes that are suggested by the statistical mechanics of self avoiding paths on the pre sierpinski gasket we prove that there are three types of continuum limit processes according to the parameters defining the statistical mechanics of self avoiding paths i the self avoiding process we construct in this paper ii a deterministic motion along a peano curve on the finite sierpinski gasket iii a deterministic motion along a line segment

multifractality tests using bootstrapped wavelet leaders multifractal analysis which mostly consists of measuring scaling exponents is becoming a standard technique available in most empirical data analysis toolboxes making use of the most recent theoretical results it is based here on the estimation of the cumulants of the log of the wavelet leaders an elaboration on the wavelet coefficients these log cumulants theoretically enable discrimination between mono and multifractal processes as well as between simple log normal multifractal models and more advanced ones the goal of the present contribution is to design nonparametric bootstrap hypothesis tests aiming at testing the nature of the multifractal properties of stochastic processes and empirical data bootstrap issues together with six declinations of test designs are analyzed their statistical performance significances powers and p values are assessed and compared by means of monte carlo simulations performed on synthetic stochastic processes whose multifractal properties and log cumulants are known theoretically a priori we demonstrate that the joint use of wavelet leaders log cumulants and bootstrap procedures enable us to obtain a powerful tool for testing the multifractal properties of data this tool is practically effective and can be applied to a single observation of data with finite length

self similar random measures are locally scale invariant in an earlier paper patzschke and u zahle 11 have proved the existence of a fractional tangent measure at the typical point of a self similar random measure under rather special technical assumptions in the present paper we remove the most restrictive one here we suppose the open set condition for the similarities a constant positive lower bound for the random contraction ratios and vanishing on the boundary of the open set with probability 1 the tangent measure isd scale invariant whered is the similarity dimension of moreover we approximate the tangential distribution by means of and use this in order to prove that the hausdorff dimension of the tangent measure equalsd since the former coincides with the hausdorff dimension of we obtain an earlier result of mauldin and williams 9 as a corollary

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

dimensions and measures in infinite iterated function systems the hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated conformal measures for such systems reflecting geometric properties of the limit set are introduced proved to exist and to be unique the existence of a unique invariant probability equivalent to the conformal measure is derived our methods employ the concepts of the perron frobenius operator symbolic dynamics on a shift space with an infinite alphabet and the properties of the above mentioned ergodic invariant measure a formula for the hausdorff dimension of the limit set in terms of the pressure function is derived fractal phenomena not exhibited by finite systems are shown to appear in the infinite case in particular a variety of conditions are provided for hausdorff and packing measures to be positive or finite and a number of examples are described showing the appearance of various possible combinations for these quantities one example given special attention is the limit set associated to the complex continued fraction expansion in particular lower and upper estimates for its hausdorff dimension are given a large natural class of systems whose limit sets are dimensionless in the restricted sense is described

new multiplier methods for summing classical eigenfunction expansions on unifie et on generalise un certain nombre de resultats en theorie de sommabilite classique pour des developpements de sturm liouville regulier et singulier

a cascade decomposition theory with applications to markov and exchangeable cascades a multiplicative random cascade refers to a positive t martingale in the sense of kahane on the ultrametric space t 0 1 b 1 a new approach to the study of multiplicative cascades is introduced the methods apply broadly to the problems of i non degeneracy criterion ii dimension spectra of carrying sets and iii divergence of moments criterion specific applications are given to cascades generated by markov and exchangeable processes as well as to homogeneous independent cascades 1 positive t martingales positive t martingales were introduced by jean pierre kahane as the general framework for independent multiplicative cascades and random coverings although originating in statistical theories of turbulence the general framework also includes certain spin glass and random polymer models as well as various other spatial distributions of interest in both probability theory and the physical sciences for basic definitions let t be a compact metric space with borel sigmafield b and let f p be a probability space together with an increasing sequence fn n 1 2 of sub sigmafields of f a positive t martingale is a sequence qn of b f measurable non negative functions on t such that i for each t t qn t n 0 1 is a martingale adapted to fn n 0 1 ii for p a s qn n 0 1 is a sequence of borel measurable non negative real valued functions on t let m t denote the space of positive borel measures on t and suppose that qn t is a positive t martingale for m t such that q t eqn t l let n qn denote the random measure defined by qn and dqn d t qn t t t then essentially by the martingale convergence theorem one obtains a random borel measure q such that for f c t 1 1 lim n t f t qn t dt t f t q dt a s received by the editors august 18 1994 1991 mathematics subject classification primary 60g57 60g30 60g42 secondary 60k35 60d05 60j10 60g09

an improved multifractal formalism and self similar measures abstract to characterize the geometry of a measure its generalized dimensions dq have been introduced recently the mathematically precise definition given by falconer fractal geometry 1990 turns out to be unsatisfactory for reasons of convergence as well as of undesired sensitivity to the particular choice of coordinates in the negative q range a new definition is introduced which is also based on box counting but which carries relevant information about for negative q also in particular the legendre connection between generalized dimensions and the multifractal spectrum is established using large deviations and rigorous proofs are provided for the implicit formula giving the generalized dimensions of selfsimilar measures which was until now known only for positive q

two guidelines for bootstrap hypothesis testing two guidelines for nonparametric bootstrap hypothesis testing are highlighted the first recommends that resampling be done in a way that reflects the null hypothesis even when the true hypothesis is distant from the null the second guideline argues that bootstrap hypothesis tests should employ methods that are already recognized as having good features in the closely related problem of confidence interval construction violation of the first guideline can seriously reduce the power of a test sometimes this reduction is spectacular since it is most serious when the null hypothesis is grossly in error the second guideline is of some importance when the conclusion of a test is equivocal it has no direct bearing on power but improves the level accuracy of a test

on the asymptotics of the eigenvalue counting function for random recursive sierpinski gaskets we consider natural laplace operators on random recursive affine nested fractals based on the sierpinski gasket and prove an analogue of weyl s classical result on their eigenvalue asymptotics the eigenvalue counting function n is shown to be of order ds 2 as where we can explicitly compute the spectral dimension d s moreover the limit n ds 2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable this random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal