measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

self similarity and network perspective of the chinese fund market abstract by testing 88 different funds of the chinese fund market cfm we find fractal behavior and long range correlations in the return series which are insensitive to the kind of funds meanwhile a power law relationship between the deviation d of prices and the hurst exponent h has been obtained which may be useful for predicting the price time series in addition with funds being viewed as nodes and the connections among the funds being determined by the cross correlation coefficients using a winner takes all approach we investigate the topological properties of the fund network our analysis reveals that during different time periods the cumulative degree distributions of the fund network all obey the double power law format moreover the small world property is also found for the fund network

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

random multifractals negative dimensions and the resulting limitations of the thermodynamic formalism the story described in this paper has started with the death or survival criterion which the author published in 1972 1974 and had obtained in 1968 while investigating kolmogorov s hypothesis that the turbulent dissipation epsilon dx in a box is log normally distributed using this criterion the present paper discusses the concrete significance of negative fractal dimensions they arise in those random multifractal measures for which the cramer function f alpha the spectrum of singularities satisfies f alpha 0 and concerns three aspects of such measures a the fine grained multifractal properties which are non random and the same for almost all realizations b the properties obtained by using the partition function formalism and c the typical coarse grained multifractal properties however the f alpha s in the range where f alpha 0 say nothing about the variability of coarse grained properties between samples a description of these fluctuations hence a fuller multifractal description of the measure is shown to be provided by the values of f alpha in the range where f alpha 0 we prefer to reserve the term thermodynamic for the fine grained and partition functional properties and to say that the coarse grained properties go beyond the thermodynamics i e are not macroscopic but mesoscopic

self similarity degree of deformed statistical ensembles abstract we consider self similar statistical ensembles with the phase space whose volume is invariant under the deformation that squeezes expands the coordinate and expands squeezes the momentum the related probability distribution function is shown to possess a discrete symmetry with respect to manifold action of the jackson derivative to be a homogeneous function with a self similarity degree q fixed by the condition of invariance under n 1 fold action of the related dilatation operator in slightly deformed phase space we find the homogeneous function is defined with the linear dependence at n 0 whereas the self similarity degree equals the gold mean at n 1 and q n in the limit n dilatation of the homogeneous function is shown to decrease the self similarity degree q at n 0

detrending moving average algorithm for multifractals the detrending moving average dma algorithm is a widely used technique to quantify the long term correlations of nonstationary time series and the long range correlations of fractal surfaces which contains a parameter ensuremath theta determining the position of the detrending window we develop multifractal detrending moving average mfdma algorithms for the analysis of one dimensional multifractal measures and higher dimensional multifractals which is a generalization of the dma method the performance of the one dimensional and two dimensional mfdma methods is investigated using synthetic multifractal measures with analytical solutions for backward ensuremath theta 0 centered ensuremath theta 0 5 and forward ensuremath theta 1 detrending windows we find that the estimated multifractal scaling exponent ensuremath tau q and the singularity spectrum f ensuremath alpha are in good agreement with the theoretical values in addition the backward mfdma method has the best performance which provides the most accurate estimates of the scaling exponents with lowest error bars while the centered mfdma method has the worse performance it is found that the backward mfdma algorithm also outperforms the multifractal detrended fluctuation analysis the one dimensional backward mfdma method is applied to analyzing the time series of shanghai stock exchange composite index and its multifractal nature is confirmed

sources sinks and wavenumber selection in coupled cgl equations and experimental implications for counter propagating wave systems we study the coupled complex ginzburg landau cgl equations for traveling wave systems and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems we focus on the regime in which sources and sinks separate patches of left and right traveling waves i e the case that these modes suppress each other we present in detail the framework to analyze these coherent structures and show that the theory predicts a number of general properties which can be tested directly in experiments our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations there generally exists a symmetric stationary source solution which sends out waves with a unique frequency and wave number sinks on the other hand occur in two parameter families and play an essentially passive role being sandwiched between the sources these simple but general results imply that sources are important in organizing the dynamics of the coupled cgl equations simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2d complex ginzburg landau equations sources can send out stable waves convectively unstable waves or absolutely unstable waves we show that there exists an additional dynamical regime where both single and bimodal states are unstable the ensuing chaotic states have no counterpart in single amplitude equations a third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate e this is related to the fact that the standard coupled cgl equations are not uniform in e in particular when the group velocity term dominates over the linear growth term no stationary source can exist however sources displaying nontrivial dynamics can often survive here our results for the existence multiplicity wavelength selection dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments we therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment in addition they bring up a large number of new research issues and open problems which are listed explicitly in the concluding section

complex fractal dimensions describe the hierarchical structure of diffusion limited aggregate clusters we analyze large diffusion limited aggregates and uncover a ital discrete scaling invariance in their inner structure which can be quantified by the introduction of a set of ital complex fractal dimensions we provide a theoretical framework and prediction of their values based on renormalization group theory and a previous wavelet analysis copyright ital 1996 the american physical society

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

weighted sensitivity minimization general plants in h and rational weights abstract for general plants in h and rational weighting functions the problem of computing the weighted h optimal sensitivity is considered a general formula is derived for a function whose singularities determine the optimal sensitivity 6 a determinantal expression for the optimal sensitivity is then deduced and related to a conjecture of zhou and khargonekar

asymptotic expansions for fourier transform of singular self affine measures abstract singular continuous self affine measures invariant with respect to some stochastic dynamical systems generated by a finite set of affine contractions on the axis are studied the hausdorff dimension of their support and leading terms of l 2 norm asymptotic of the corresponding cut off characteristic function are computed it is shown that the power law behavior multiplied by some oscillating functions of a logarithm of a cut off parameter for these terms is typical the inverse reconstruction problem of the invariant measure from the high frequency behavior of its characteristic function is discussed

generalized q analysis of log periodicity applications to critical ruptures we introduce a generalization of the q analysis which provides a nonparametric tool for the description and detection of log periodic structures associated with discrete scale invariance we use this generalized q analysis to construct a signature called the h q derivative of discrete scale invariance which we use to detect the log periodicity in the cumulative energy release preceding the rupture of five pressure tanks made of composite carbon matrix material we investigate the significance level of the spectral lomb periodogram of the optimal h q derivative we confirm and strengthen previous parametric results that the cumulative energy release exhibits log periodicity before rupture however our tests to use this method as a scheme for the prediction of the critical value of the stress at rupture are not encouraging

linear control theory with an h 0e optimality criterion this expository paper sets out the principal results in bf h infty control theory in the context of continuous time linear systems the focus is on the mathematical theory rather than computational methods

singular values and vectors of a class of hankel operators abstract this paper considers the problem of determining the singular values and vectors of the class of hankel operators whose symbol can be written in the form w where is inner and w is rational we will obtain a new and simplified proof of the fact conjectured in 17 and first proved in 9 that the zeros of a certain determinantal expression coincide with the hankel singular values we will also show how this expression can be used to generate the singular vectors

weighted sensitivity minimization for delay systems in this note we discuss the h infty sensitivity minimization problem for linear time invariant delay systems while the unweighted case reduces to simple nevanlinna pick interpolation the weighted case turns out to be much more complicated and demands certain functional analytic techniques for its solution

microscopic simulation of limit cycle behavior in spatially extended systems the onset of homogeneous oscillations in spatially extended system is considered the master equation formulation shows that in a one dimensional system there exists a finite length beyond which the homogeneous oscillations are destroyed microscopic simulations are used to investigate the status of this prediction and quantitative agreement is obtained the origin of the desynchronization mechanism is clarified

on the parametrization of the suboptimal solutions in generalized interpolation abstract we apply methods previously developed to give a explicit parametrization of the suboptimal solutions to the standard h generalized interpolation problem this gives in particular a simple parametrization of all the suboptimal controllers of the standard sensitivity minimization problem

multiscale multifractal analysis of heart rate variability recordings with a large number of occurrences of arrhythmia human heart rate variability in the form of time series of intervals between heart beats shows complex fractal properties recently it was demonstrated many times that the fractal properties vary from point to point along the series leading to multifractality in this paper we concentrate not only on the fact that the human heart rate has multifractal properties but also that these properties depend on the time scale in which the multifractality is measured this time scale is related to the frequency band of the signal we find that human heart rate variability appears to be far more complex than hitherto reported in the studies using a fixed time scale we introduce a method called multiscale multifractal analysis mma which allows us to extend the description of heart rate variability to include the dependence on the magnitude of the variability and time scale or frequency band mma is relatively immune to additive noise and nonstationarity including the nonstationarity due to inclusions into the time series of events of a different dynamics e g arrhythmic events in sinus rhythm the mma method may provide new ways of measuring the nonlinearity of a signal and it may help to develop new methods of medical diagnostics

multifractals embedded in short time series an unbiased estimation of probability moment an exact estimation of probability moments is the base for several essential concepts such as the multifractals the tsallis entropy and the transfer entropy by means of approximation theory we propose a new method called factorial moment based estimation of probability moments theoretical prediction and computational results show that it can provide us an unbiased estimation of the probability moments of continuous order calculations on probability redistribution model verify that it can extract exactly multifractal behaviors from several hundred recordings its powerfulness in monitoring evolution of scaling behaviors is exemplified by two empirical cases i e the gait time series for fast normal and slow trials of a healthy volunteer and the closing price series for shanghai stock market by using short time series with several hundred lengths a comparison with the well established tools displays significant advantages of its performance over the other methods the factorial moment based estimation can evaluate correctly the scaling behaviors in a scale range about three generations wider than the multifractal detrended fluctuation analysis and the basic estimation the estimation of partition function given by the wavelet transform modulus maxima has unacceptable fluctuations besides the scaling invariance focused in the present paper the proposed factorial moment of continuous order can find its various uses such as finding nonextensive behaviors of a complex system and reconstructing the causality relationship network between elements of a complex system

differential flow instability in the ginzburg landau and swift hohenberg approximations abstract the effects of a differential flow of the components of a reaction diffusion system which is close to its stability boundary are described within the long wavelength approximation in the vicinity of the hopf bifurcation the system s evolution is governed by a complex ginzburg landau equation modified by a purely imaginary convective term if the system is near the zero real eigenvalue bifurcation the governing equation is a modified swift hohenberg equation in both cases the homogeneous stable reference steady state may be destabilized by the differential flow in the ginzburg landau equation the destabilization occurs as long as the flow velocity exceeds some critical value v cr which tends to zero as the system approaches the hopf bifurcation in the modified swift hohenberg equation the flow has either a destabilizing or stabilizing effect depending on the sign of one of the system parameters destabilization occurs when the flow velocity exceeds some threshold however in this case the threshold remains finite even at the bifurcation point in both ginzburg landau and swift hohenberg equations the differential flow instability produces traveling plane waves the stability analysis shows that once a periodic plane wave is established its spatial period remains unchanged over a finite range of the flow velocity and changes in discrete steps the phenomenon of wavenumber locking wavenumber locking is verified in numerical experiments with the ginzburg landau equation near hopf bifurcation the benjamin feir instability may occur in this case irregular traveling waves are found but a regular component of the wave pattern survives depending on a parameter the differential flow either promotes or deters the benjamin feir instability as a result the increasing flow may swithc the periodic wave pattern into a irregular state or conversely may stabilize the previously induced irregular pattern and produce periodic waves

negative fractal dimensions and multifractals abstract a new notion of fractal dimension is defined when it is positive it effectively falls back on known definitions but its motivating virtue is that it can take negative values which measure usefully the degree of emptiness of empty sets the main use concerns random multifractals for which f

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

on decoupling the h optimal sensitivity problem for products of plants abstract in this note we show how to solve the h optimal sensitivity problem for a siso plant p s p 1 s p 2 s given the solutions for p 1 s p 2 s this allows us to solve the problem for systems of the form e hs p 0 s where p 0 s is the transfer function of a stable lti finite dimensional system

on the properties of random multiplicative measures with the multipliers exponentially distributed under the formalism of annealed averaging of the partition function a type of random multifractal measures with the multipliers exponentially distributed is investigated in detail branching emerges in the curve of generalized dimensions and negative values of generalized dimensions arise three equivalent methods of classification of the random multifractal measures are proposed which is based on i the properties of the curves of generalized dimensions ii the solution properties of equation q 0 and iii the relative position of the curve f and the diagonal f in the first quadrant these classes of measures correspond to 0 1 0 1 1 and 0 1 0 respectively phase diagram is introduced to illustrate the diverse performance of the random measures that is multiplicatively generated

new anomalous multiplicative multifractals left sided ƒ α and the modelling of dla abstract a sharp distinction is drawn between general multiplicative multifractals as originally introduced by the author and the more familiar but more restricted class defined by frisch and parisi and by halsey et al all the general multiplicative multifractals are exactly renormalizable by design however this is important and perhaps suprising it is shown that the scaling relations that serve to define the restricted class fail to extend to certain multiplicative multifractals as a result in addition to the familiar restricted class the general multifractals are shown to allow for a wide variety of anomalous behaviors several of which are described we believe that suitable examples of these anomalies are of the most important concrete applications to fully developed turbulence and to diffusion limited aggregates dla this paper concerns several new classes of multifractals we have recently proposed as tools to model the anomalies of dla they are left sided that is characterized by max with max d 0 related multifractals characterized by max 0 and even by max and min 0 are encountered along the way

stationary self propagating fronts in potential flow abstract we analyze the problem of stationary self propagating fronts in potential flow the issues of local existence and uniqueness for solutions of the ode describing stationary fronts multiplicity of solutions and linearized stability of a stationary front as a solution of the hyperbolic evolution equation are addressed the results are illustrated in the case of the dual source system which is a simple model of a combustion system in which local extinction may arise model extensions for combustion applications are presented

secondary structures in a one dimensional complex ginzburg landau equation with homogeneous boundary conditions experiments in extended systems such as the counter rotating couette taylor flow or the taylor dean flow system have shown that patterns with vanishing amplitude may exhibit periodic spatio temporal defects for some range of control parameters these observations could not be interpreted by the complex ginzburg landau equation cgle with periodic boundary conditions we have investigated the one dimensional cgle with homogeneous boundary conditions we found that in the benjamin feir stable region the basic wave train bifurcates to state with periodic spatio temporal defects the numerical results match the observations quite well we have built a new state diagram in the parameter plane spanned by the criticality or equivalently the linear group velocity and the nonlinear frequency detuning

some remarks on optimal interpolation abstract we give in this note very simple new formula for the optimal solution of a generalized interpolation problem which arises in the weighted sensitivity h minimization for distributed systems this is based on our work on skew toeplitz operators from 2

approximation of stable systems by laguerre filters laguerre methods are used to develop concrete techniques and results for approximating stable infinite dimensional systems both hankel norm and l approximation by laguerre filters are studied the infimum of the hankel norm approximation error when the system is approximated by laguerre filters of given maximum order is given by the norm of a certain shifted hankel operator and the best laguerre filter to approximate the system is obtained essentially from the associated eigenproblem for the shifted hankel operator hankel optimal laguerre filters are here called cf laguerre filters as there is a certain connection to the caratheodory fejer interpolation problem and typically provide near best solutions to related l approximation problems a time domain compressed laguerre shift method for solving certain nehari problems for delay systems is used to determine analytically the achievable hankel error norms and thus also to obtain bounds for the achievable l error norm for delay systems a result on a laguerre filter approximation problem in the hilbert schmidt norm is given as well

time periodic spatial chaos in the complex ginzburg landau equation the phenomenon of time periodic evolution of spatial chaos is investigated in the frames of one and two dimensional complex ginzburg landau equations it is found that there exists a region of the parameters in which disordered spatial distribution of the field behaves periodically in time the boundaries of this region are determined the transition to the regime of spatiotemporal chaos is investigated and the possibility of describing spatial disorder by a system of ordinary differential equations is analyzed the effect of the size of the system on the shape and period of oscillations is investigated it is found that in the two dimensional case the regime of time periodic spatial disorder arises only in a narrow strip the critical width of which is estimated the phenomenon investigated in this paper indicates that a family of limit cycles with finite basins exists in the functional phase space of the complex ginzburg landau equation in finite regions of the parameters

the structure of spiral domain patterns and shocks in the 2d complex ginzburg landau equation abstract cellular patterns appear spontaneously in a number of nonequilibrium systems governed by the dynamics of a complex field in the case of the complex ginzburg landau equation disordered cells of effectively frozen spirals appear separated by thin walls shocks on a scale much larger than the basic wavelength of the spirals we show that these structures can be understood in very simple terms in particular we show that the walls are to a good approximation segments of hyperbolae and this allows us to construct the wall pattern given the vortex centers and a phase constant for each vortex the fact that the phase is only defined up to an integer multiple of 2 introduces a quantization condition on the sizes of the smallest spiral domains the transverse structure of the walls is analyzed by treating them as heteroclinic connections of a system of ordinary differential equations the structure depends on the angle the wall makes with the local phase contours and the behavior can be either monotonic or oscillatory depending on the parameters

stability limits of traveling waves and the transition to spatiotemporal chaos in the complex ginzburg landau equation abstract due to the convective nature of the eckhaus instability in oscillatory media there exists a conventively unstable wavenumber band in the benjamin feir unstable region this appears to explain at least qualitatively some features of spatiotemporal chaos found recently in the ginzburg landau theory

two results on entire solutions of ginzburg landau system in higher dimensions in this short article we prove two results on the ginzburg landau system of equations u u u 2 1 where u c2 rn rm n m 1 first we prove a liouville type theorem which asserts that every solution u satisfying rn u 2 1 2 is constant and of unit norm provided n 4 here m 1 in our second result we give an answer to a question raised by brezis open problem 3 of proceedings of the symposium on pure mathematics vol 65 american mathematical society providence ri 1999 about the symmetry for the ginzburg landau system in the case n m 3 we also formulate three open problems concerning the classification of entire solutions of the ginzburg landau system in any dimension

on the h optimal sensitivity problem for systems with delays in this paper we extend some of the results of ieee trans automat control ac 31 1986 pp 763 766 to more general delay systems in particular we analyze the effect of the interaction of delays and nonminimum phase zeros on the h infty optimal weighted sensitivity

dynamics of spiral waves in non equilibrium media in this paper the spiral waves in two dimensional nonequilibrium media are studied on the basis of the two dimensional ginzburg landau equation in the general case the perturbation of origin spiral waves leads to the spiral rotation and to the motion of spiral cores motion in media with hard and soft excitation is analysed separately

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

on the nehari problem for a certain class of l functions appearing in control theory ii abstract in this paper we generalize the results of c foias and a tannenbaum j functional analysis 74 1987 146 159 on the classical nehari problem to a much broader class of operators we relate our results to the computation of the discrete spectrum for a certain class of contractions which appear frequently in control engineering applications

on h optimal sensitivity theory for siso feedback systems this paper deals with the design of feedback controllers which minimize the h infty norm of the sensitivity function suitably weighted this approach to the theory of feedback design was introduced by zames 1 and developed by zames and francis 2 in this paper the theory of sarason 3 is applied to the determination of the optimal weighted sensitivity function and an upper bound on its norm the problem of achieving small sensitivity over a specified frequency band is studied and the effect of nonminimum phase is elucidated finally a method is introduced for handling plant poles and zeros on the imaginary axis

primary and secondary noninear global instability we study oscillating solutions of the complex ginzburg landau equation in a semi infinite domain asymptotic to a saturated traveling wave at c1 and subject to a homogeneous upstream boundary condition at x d 0 this inlet condition breaks the galilean invariance giving the advection velocity the role of a control parameter we give a criterion for the existence of a nontrivial solution called a nonlinear global mode or a self sustained resonance and we obtain the selected frequency the threshold and the frequency are shown to be determined by the linear absolute instability transition we undertake a singular perturbation analysis which first proves the validity of the criterion and secondly yields scaling laws for the frequency the growth length and the slope at the origin of the nonlinear global modes as functions of the criticality parameter comparisons with direct numerical simulation of the ginzburg landau model validate these predictions furthermore the results are in excellent agreement with numerical simulation of the taylor couette problem with throughflow by buchel et al p buchel m lucke d roth r schmitz pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem taylor vortices in axial flow phys rev e 53 1996 4764 4777 and of rayleigh benard flow with an added poiseuille flow by muller et al h w muller m lucke m kamps convective patterns in horizontal flow europhys lett 10 1989 451 456 h w muller m lucke m kamps transversal convection patterns in horizontal shear flow phys rev a 45 1992 3714 3726 the numerical simulations indicate that the nonlinear global modes are stable if the saturated traveling wave to which the nonlinear global mode is asymptotic i e the asymptotic downstream part of the global modes is not absolutely unstable to perturbations a complete analysis of the dispersion relation of the secondary instability of saturated traveling waves is given 1999 elsevier science b v all rights reserved

on hankel operators associated with a class of non toeplitz operators the classical nehari theorem 7 p 43 gives the distance from a bounded measurable function cp on the unit circle t to h the algebra of bounded measurable functions on t with zero negative fourier coefficients the distance is the norm of the hankel operator defined by cp this result was followed by the result of hartman 7 p 61 giving the distance from cp to the algebra h c generated by h and the continuous functions on t as the essential norm of the hankel operator defined by rp as was shown by adamjan arov and krein 11 these two results are extremal cases of the following result if r denotes the rational functions on t with no more than n poles counting multiplicities in the interior of t then the distance from cp to h r is given by the nth s number of the hankel operator defined by cp in this paper we present operator theoretic analogues of these results we define for an operator t acting on the hilbert sequence h2 of square summable complex sequences a sequence of operators on h which when it converges converges to a hankel operator h t in the case where t is a toeplitz operator h t is the classical hankel operator associated with t we give a necessary and sufficient condition for the existence of h t and show that h t exists for a large class of non toeplitz operators we then define the corresponding operator theoretic analogues to the above function theoretic results in terms of the s numbers of the hankel operator h t associated with t note that since h t is in fact a hankel operator in the classical sense all these numbers can be computed using the algorithms of foias tannenbaum s a problem closely related to the problem discussed here was studied by barria and halmos s the relationship between the two problems is discussed in detail in 9 1 oc22 1236190 3 00

stabilization and sensitivity for eventually time invariant systems abstract this paper studies the stability and minimal sensitivity of a time varying system which as time passes becomes more and more time invariant

numerical investigations of discrete scale invariance in fractals and multifractal measures fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems here we focus on lattice multifractals which exhibit complex exponents associated with observable log periodicity we perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments a novel numerical approach is proposed to extract the log frequencies in the non lattice case there is no visible log periodicity i e no preferred scaling ratio since the set of complex exponents spreads irregularly within the complex plane a non lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non lattice one an algorithm for the construction of the lattice sequence is proposed explicitly

synchronized disorder in a 2d complex ginzburg landau equation the phenomenon of spiral pair synchronization by oscillating strips nozaki bekki solution in a 2d cgle is investigated analytically the equations describing the interaction of the strips with one another and with spirals are derived analysis of the equations shows that under certain conditions the strips lead to frequency and phase locking of the spirals in this case the spiral pair dipole is aligned parallel to the strips with the position along the strips being arbitrary thus the interaction with strips may transform the spatio temporal chaos of spirals to the regime of periodically oscillating spatial disorder the dynamics of circular strips is investigated and their lifetime is estimated the behavior of the spirals bounded by circular stript is analysed

sensitivity minimization for arbitrary siso distributed plants abstract in this note we write down an explicit formula for the computation of the optimal sensitivity of an arbitrary single input single output linear invariant distributed plant we moreover discuss the uniqueness of the optimal compensator and work out a design example

on the weighted sensitivity minimization problem for delay systems we consider the h optimal sensitivity problem for delay systems in particular we consider computation of inf w q q h jr where w s is any function in rh jr and in h jr is any inner function we derive a new explicit solution in the pure delay case where e sh h 0

an interpolation problem associated with h optimal design in systems with distributed input lags the symbol for infinity appears as superscript in title where indicated by the word infinity in brackets

nehari problems and equalizing vectors for infinite dimensional systems for a class of infinite dimensional systems we obtain a simple frequency domain solution for the suboptimal nehari extension problem the approach is via j spectral factorization and it uses the concept of an equalizing vector moreover the connection between the equalizing vectors and the nehari extension problem is given

vector valued multifractal measures we propose a framework for vector valued measures which permits vector analogues if the singular measures which arise in multifractal theory we obtain some of their properties and calculate the multifractal spectrum and l p dimensions of certain self similar vector valued measures

evidence of intermittent cascades from discrete hierarchical dissipation in turbulence abstract we present the results of a search of log periodic corrections to scaling in the moments of the square of velocity increments taken as proxies of the energy dissipation rate in experiments at high reynolds number 2500 of three dimensional fully developed turbulence a simple dynamical representation of the richardson kolmogorov cartoon of a cascade shows that standard averaging techniques erase by their very construction the possible existence of log periodic corrections to scaling associated with a discrete hierarchy to remedy this drawback we introduce a novel canonical averaging that we test extensively on synthetic examples constructed to mimic the interplay between a weak log periodic component and rather strong multiplicative and phase noises tests are also presented on 1 f5 3 random phase noise on the multifractal random walk model and on lattice and non lattice multifractal models our extensive tests confirm the remarkable observation of statistically significant log periodic corrections to scaling with a preferred scaling ratio for length scales compatible with the value 2 a strong confirmation of this result is provided by the identification of up to five harmonics of the fundamental log periodic undulations associated with up to five levels of the underlying hierarchical dynamical structure a natural interpretation of our results is that the richardson kolmogorov mental picture of a cascade becomes a realistic description if one allows for intermittent births and deaths of discrete cascades at varying scales

particle aggregation versus cluster aggregation in high dimensions we distinguish two different types of irreversible aggregation accretion of individual particles and successive aggregation of clusters of comparable size in aggregation of particles which follow trajectories of fractal dimensiond 1 we show that physical limits on the aggregation rate impose a lower bound on the fractal dimensiond 0 of the aggregate ind dimensional space d 0 d d 1 1 thus aggregation of ballistic particles withd 1 1 is not fractal by contrast cluster aggregates appear to attain a finite limitingd 0 in high dimensions we present a soluble model with this property and argue that it should agree with sutherland s binary aggregation model in high dimensions for this model d 0 depends continuously on a parameter the exponent is not universal

nonlinear analysis of the eckhaus instability modulated amplitude waves and phase chaos with nonzero average phase gradient abstract we analyze the eckhaus instability of plane waves in the one dimensional complex ginzburg landau equation cgle and describe the nonlinear effects arising in the eckhaus unstable regime modulated amplitude waves maws are quasi periodic solutions of the cgle that emerge near the eckhaus instability of plane waves and cease to exist due to saddle node sn bifurcations these maws can be characterized by their average phase gradient and by the spatial period p of the periodic amplitude modulation a numerical bifurcation analysis reveals the existence and stability properties of maws with arbitrary and p maws are found to be stable for large enough and intermediate values of p for different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude defects form from perturbed plane waves for parameter values above the sn of the corresponding maws the break down of phase chaos with average phase gradient 0 wound up phase chaos is thus related to these sns a lower bound for the break down of wound up phase chaos is given by the necessary presence of sns and an upper bound by the absence of the splitting instability of maws

a polynomial approach to hankel norm and balanced approximations abstract a new unified approach to problems of hankel norm and balanced approximations is presented which is based on a combination of polynomial algebra and the geometry of invariant subspaces contrary to state space methods where contact with external properties of systems is indirect the approach presented yields new insights into basic properties of hankel norm approximation and balanced realizations several approximation results are interpreted geometrically in terms of projections also duality results in this area are printed out

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

analytic system problems and j lossless coprime factorization for infinite dimensional linear systems abstract this paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an h norm bound to a class of systems with irrational transfer matrices using the coprime factorization description the h control problem can be reduced to two stable analytic system problems such problems have solutions if and only if a certain j lossless factorization exists the full h synthesis problem is shown to be equivalent to the solution of two nested j lossless factorizations if the irrational transfer matrix has a state space realization then the known state space formulas for the h control problem may be recovered using the relationship between j lossless factorizations and solutions of riccati equations however the results derived here are valid for a larger class of infinite dimensional systems

spatiotemporal chaos in the one dimensional complex ginzburg landau equation the dynamical behavior of a large one dimensional system obeying the cubic complex ginzburg landau equation is studied numerically as a function of parameters near a supercritical bifurcation two types of chaotic behavior can be distinguished beyond the benjamin feir instability a phase turbulence regime with a conserved phase winding number and no phase dislocations space time defects and a defect regime with a nonzero density of defects the transition between the two can either be continuous or discontinuous hysteretic depending on parameters the spatial decay of the phase correlation function is inferred to be exponential in both regimes with a sharp decrease of the correlation length upon entering the defect phase the temporal decay of correlations is exponential in the defect regime

the suboptimal nehari problem for well posed linear systems we solve the suboptimal nehari problem for a transfer function that has a state space realization as a system stable input output and input output stable well posed linear system we obtain an explicit solution in terms of the state space parameters