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

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

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

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

the structure of as gorenstein algebras abstract in this paper we define a notion of as gorenstein algebra for n graded algebras and show that symmetric as regular algebras of gorenstein parameter 1 are exactly preprojective algebras of quasi fano algebras this result can be compared with the fact that symmetric graded frobenius algebras of gorenstein parameter 1 are exactly trivial extensions of finite dimensional algebras the results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite dimensional algebras

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

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

bifurcations of tori and phase locking in a dissipative system of differential equations abstract in a system of ordinary differential equations obtained through a seven mode truncation of the plane incompressible navier stokes equations a two dimensional torus undergoes first two period doubling bifurcations and then a transition to a strange attractor this strange attractor of liapunov dimension larger than three in a wide parameter interval is characterized by a power spectrum which retains the two fundamental frequencies of the original torus superimposed on a broad jagged background as the liapunov dimension goes down towards two an interesting phenomenon of phase locking occurs which gives rise to an alternation of chaotic and periodic behavior

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

on the use of rootfinding ode software for the solution of a common problem in nonlinear dynamical systems we discuss how rootfinding which is built into some ode software can be used to generate poincare sections an interactive program is also described

helices on del pezzo surfaces and tilting calabi yau algebras we study tilting for a class of calabi yau algebras associated to helices on fano varieties we do this by relating the tilting operation to mutations of exceptional collections for helices on del pezzo surfaces the algebras are of dimension three and using an argument of herzog together with results of kuleshov and orlov we obtain a complete description of the tilting process in terms of quiver mutations

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

torus breakdown in noninvertible maps we propose a criterion for the destruction of a two dimensional torus through the formation of an infinite set of cusp points on the closed invariant curves defining the resonance torus this mechanism is specific to noninvertible maps the cusp points arise when the tangent to the torus at the point of intersection with the critical curve l 0 coincides with the eigendirection corresponding to vanishing eigenvalue for the noninvertible map further parameter changes lead typically to the generation of loops self intersections of the invariant manifolds followed by the transformation of the torus into a complex chaotic set

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

computation continuation and bifurcation of torus solutions for dissipative maps and ordinary differential equations abstract a periodic solution can be considered as a fixed point of a poincare map we shall demonstrate how we can generalize this fixed point equivalence to quasi periodic solutions with two periods thus the quasi periodic solution or bi periodic solution will be a fixed point of a more general poincare map the stability of the bi periodic solution is then equivalent to the stability of the fixed point we demonstrate how we can follow the fixed point in dependence of any control parameter in the system furthermore we demonstrate both how the bi periodic solution may bifurcate to more complicated solutions and how it may arise as a bifurcation from a periodic solution our work is algorithmic in spirit and our statements are based on numerical evidence

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

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

low frequency behavior in a frictionally excited beam abstract a cantilevered beam excited by a periodically reciprocating friction contact surface exhibited extremely low frequency responses with frequencies as low as 100 times lower than the driver example responses near 1 1 and 1 2 resonances between the excitation and the degree of freedom normal to the direction of sliding show two frequency quasiperiodicity and in one case three frequency quasiperiodicity underlying circle maps were extracted and winding numbers were matched to response frequencies quantified in the fast fourier transforms of the responses a torus doubling bifurcation was documented

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

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

n representation infinite algebras abstract from the viewpoint of higher dimensional auslander reiten theory we introduce a new class of finite dimensional algebras of global dimension n which we call n representation infinite they are a certain analog of representation infinite hereditary algebras and we study three important classes of modules n preprojective n preinjective and n regular modules we observe that their homological behaviour is quite interesting for instance they provide first examples of algebras having infinite ext 1 orthogonal families of modules moreover we give general constructions of n representation infinite algebras applying minamoto s theory on fano algebras in non commutative algebraic geometry we describe the category of n regular modules in terms of the corresponding preprojective algebra then we introduce n representation tame algebras and show that the category of n regular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra this generalizes the classical description of regular modules over tame hereditary algebras as an application we show that the representation dimension of an n representation tame algebra is at least n 2

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

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

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

numerical computation of invariant circles of maps abstract a method for the numerical computation of invariant circles of maps is presented along with appropriate techniques for its implementation the method involves solution of a functional equation by discretization and newton iteration the resulting algorithm is applied to a map of the cylinder and some examples of bifurcations of invariant circles are illustrated generalization of the method to the computation of invariant circles in more than two dimensions as well as sections of invariant tori is discussed

amplitude modulated and chaotic dynamics in resonant motion of strings abstract the weakly non linear resonant motions of a stretched string to a harmonic planar excitation are investigated one mode truncation of the in plane and out of plane equation results in a two degree to freedom system which is examined by using the method of averaging and the method of integral manifolds it is shown that the averaged equations for small enough damping possess non planar constant solutions which become unstable by a hopf bifurcation the resulting limit cycle solutions correspond to amplitude modulated whirling or ballooning motions of the string there are two limit cycle branches for the averaged equations one arising due to the hopf bifurcation and the other due to a global saddle node bifurcation the two branches merge as the damping is further reduced also with variation in detuning the isolated branch exhibits period doubling bifurcations chaotic attractors and merging of attractors giving rise to rossler as well as lorentz type solutions the truncated string equations are also directly integrated it is shown that there are non planar periodic responses that bifurcate into amplitude modulated motions on a 2 torus changes in detuning result in torus doubling merging of tori and then destruction of the torus leading to chaotic amplitude modulations the results of asymptotic analysis via the method of averaging are found to be in qualitative agreement with the actual response

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

the nonlinear dynamics of the damped and driven toda chain ii fourier and lyapunov analysis of tori abstract the dynamics of a periodically damped and driven toda chain with periodic boundary conditions is investigated with the help of spectral energy bifurcation diagrams and lyapunov diagrams it can be shown that quasiperiodic motions on tori tk of dimension k 3 exist the technical problems concerning the fourier and lyapunov analysis of the motions on tori tk k 2 are addressed

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

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

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

higher apr tilting preserves n representation infiniteness abstract we show that m apr tilting preserves n representation infiniteness for 1 m n moreover we show that these tilting modules lift to tilting modules for the corresponding higher preprojective algebras which is n 1 cy algebras we also study the interplay of the two kinds of tilting modules

the quasi periodic doubling cascade in the transition to weak turbulence abstract the quasi periodic doubling cascade is shown to occur in the transition from regular to weakly turbulent behaviour in simulations of incompressible navier stokes flow on a three periodic domain special symmetries are imposed on the flow field in order to reduce the computational effort thus we can apply tools from dynamical systems theory such as continuation of periodic orbits and computation of lyapunov exponents we propose a model ode for the quasi period doubling cascade which in a limit of a perturbation parameter to zero avoids resonance related problems the cascade we observe in the simulations is then compared to the perturbed case in which resonances complicate the bifurcation scenario in particular we compare the frequency spectrum and the lyapunov exponents the perturbed model ode is shown to be in good agreement with the simulations of weak turbulence the scaling of the observed cascade is shown to resemble the unperturbed case which is directly related to the well known doubling cascade of periodic orbits

n representation finite algebras and n apr tilting we introduce the notion of n representation finiteness generalizing representationfinite hereditary algebras we establish the procedure of n apr tilting and show that it preserves n representation finiteness we give some combinatorial description of this procedure and use this to completely describe a class of n representation finite algebras called type a

silting reduction and calabi yau reduction of triangulated categories it is shown that the silting reduction t thickp of a triangulated category t with respect to a presilting subcategory p can be realised as a certain subfactor category of t and that there is a one to one correspondence between the set of pre silting subcategories of t containing p and the set of pre silting subcategories of t thickp this is analogous to a result for calabi yau reduction this result is applied to show that the amiot guo keller construction of d calabi yau triangulated categories with d cluster tilting objects takes silting reduction to calabi yau reduction and conversely calabi yau reduction lifts to silting reduction

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