quantifying signals with power law correlations a comparative study of detrended fluctuation analysis and detrended moving average techniques detrended fluctuation analysis dfa and detrended moving average dma are two scaling analysis methods designed to quantify correlations in noisy nonstationary signals we systematically study the performance of different variants of the dma method when applied to artificially generated long range power law correlated signals with an a priori known scaling exponent alpha 0 and compare them with the dfa method we find that the scaling results obtained from different variants of the dma method strongly depend on the type of the moving average filter further we investigate the optimal scaling regime where the dfa and dma methods accurately quantify the scaling exponent alpha 0 and how this regime depends on the correlations in the signal finally we develop a three dimensional representation to determine how the stability of the scaling curves obtained from the dfa and dma methods depends on the scale of analysis the order of detrending and the order of the moving average we use as well as on the type of correlations in the signal

perturbation bound of singular linear systems a classical perturbation bound in numerical linear algebra is extended to more general cases of singular linear algebraic systems and weighted linear least squares problem componentwise perturbation analysis of linear least squares problem is also derived

behavior of geodesic length functions on teichmuller space let t be the teichmuller space of marked genus g n punctured riemann surfaces with its bordification t the augmented teichmuller space of marked riemann surfaces with nodes abi77 ber74 pro vided with the wp metric t is a complete cat 0 metric space dw03 wol03 yam04 an invariant of a marked hyperbolic struc ture is the lengthof the geodesic in a free homotopy class a basic feature of teichmuller theory is the interplay of two dimensional hyperbolic geometry weil petersson wp geometry and the behavior of geodesic length functions our goal is to develop the understanding of the intrinsic local wp geometry through a study of the gradient and hessian of geodesic length functions considerations include ex pansions for the wp pairing of gradients expansions for the hessian and covariant derivative comparability models for the wp metric as well as the behavior of wp geodesics including a description of the alexandrov tangent cone at the augmentation approximations and applications for geodesics close to the augmentation are developed an application for fixed points of group actions is described bound ing configurations and functions on the hyperbolic plane is basic to our approach considerations include analyzing the orbit of a dis crete group of isometries and bounding sums of the inverse square exponential distance

dynamical model for dna sequences we address the problem of dna sequences developing a dynamical method based on the assumption that the statistical properties of dna paths are determined by the joint action of two processes one deterministic with long range correlations and the other random and ensuremath delta function correlated the generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos that are responsible for the birth of anomalous diffusion it is assumed that the deterministic process corresponds to unknown biological rules that determine the dna path whereas the noise mimics the influence of an infinite dimensional environment on the biological process under study we prove that the resulting diffusion process if the effect of the random process is neglected is an ensuremath alpha stable l evy process with 12 we also show that if the diffusion process is determined by the joint action of the deterministic and the random process the correlation effects of the deterministic dynamics are cancelled on the short range scale but show up in the long range one we denote our prescription to generate statistical sequences as the copying mistake map cmm we carry out our analysis of several dna sequences and their cmm realizations with a variety of techniques and we especially focus on a method of regression to equilibrium which we call the onsager analysis with these techniques we establish the statistical equivalence of the real dna sequences with their cmm realizations we show that long range correlations are present in exons as well as in introns but are difficult to detect since the exon dynamics is shown to be determined by the entanglement of three distinct and independent cmm s

universal constructions in teichmuller theory we study a new model tess of a universal teichmuller space which is defined to be the collection tess of all ideal tesselations of the poincare disk together with a distinguished oriented edge modulo the natural action of the modbius group mob applying arguments from the decorated teichmuller theory of punctured surfaces which was developed elsewhere we find that tess is an infinite dimensional frehet manifold with canonical global coordinates and there are a universal sympletic form on tess which generalizes the classical weil petersson kahler two forms and a group g which generalizes the classical mapping class groups acting on tess and preserving in fact tess is isomorphic to the right coset space of mob in the group homeo of all orientation preserving homeomorphisms of the circle and one sees easily that tess naturally generalizes the other well known models of a universal teichmuller space namely bers universal teichmuller space and the right coset space of mob in the subgroup of c smooth elements of homeo which is of interest in physics the latter model supports the kirillov kostant two form and we show that the universal symplectic form pulls back to this two form implicit in our discussion so far is the fact that homeo is isomorphic to tess and one is led to ask for the characterization of various families of homeomorphisms in terms of the corresponding tesselation reporting on joint work with dennis sullivan we partially solve one such problem here using the ahlfors bers theory furthermore as in the classical decorated teichmuller theory there is an associated universal decorated teichmuller space tess which is actually of central concern in this work tess is a frechet manifold on which g acts and there is a g invariant two form just above for tess there is moreover a g invariant cell decomposition of tess which generalizes the cell decompositions of the classical decorated teichmullerspaces thus truly all of the structure of the decorated teichmuller spaces of punctured surfaces which is known from previous work arises as the restriction of a corresponding universal structure on tess

zygmund vector fields hilbert transform and fourier coefficients in shear coordinates we parametrize the space cal z of zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the farey tesselation then we express the hilbert transform and the fourier coefficients of the zygmund vector fields in terms of the above parametrization by infinitesimal shear functions finally we compute the weil petersson metric on the teichm uller space of a punctured surface in terms of shears

the decorated teichmuller space of punctured surfaces a principal 5 bundle over the usual teichmuller space of ans times punctured surface is introduced the bundle is mapping class group equivariant and admits an invariant foliation several coordinatizations of the total space of the bundle are developed there is furthermore a natural cell decomposition of the bundle finally we compute the coordinate action of the mapping class group on the total space the total space is found to have a rich equivariant geometric structure we sketch some connections with arithmetic groups diophantine approximations and certain problems in plane euclidean geometry furthermore these investigations lead to an explicit scheme of integration over the moduli spaces and to the construction of a universal teichmuller space which we hope will provide a formalism for understanding some connections between the teichmuller theory the kp hierarchy and the virasoro algebra these latter applications are pursued elsewhere

the weil petersson and thurston symplectic forms we consider the weil petersson form on the teichmuller space mathscr t s of a surface s of genus at least 2 and we compute it in terms of the shearing coordinates for mathscr t s associated to a geodesic lamination on s in the corresponding expression the weil petersson form coincides with thurston s intersection form on the space of transverse cocycles for

integrated approach to the assessment of long range correlation in time series data to assess whether a given time series can be modeled by a stochastic process possessing long range correlation one usually applies one of two types of analysis methods the spectral method and the random walk analysis the first objective of this work is to show that each one of these methods used alone can be susceptible to producing false results we thus advocate an integrated approach which requires the use of both methods in a consistent fashion we provide the theoretical foundation of this approach and illustrate the main ideas using examples the second objective relates to the observation of long range anticorrelation hurst exponent hl1 2 in real world time series data the very peculiar nature of such processes is emphasized in light of the stringent condition under which such processes can occur using examples we discuss the possible factors that could contribute to the false claim of long range anticorrelations and demonstrate the particular importance of the integrated approach in this case

method for generating long range correlations for large systems we propose a new method to generate a sequence of random numbers with long range power law correlations that overcomes known difficulties associated with large systems the new method presents an improvement on the commonly used methods we apply the algorithm to generate enhanced diffusion isotropic and anisotropic self affine surfaces and isotropic and anisotropic correlated percolation

some topics on weighted moore penrose inverse weighted least squares and weighted regularized tikhonov problems it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of linear systems the classical normwise relative condition number formulas are given by higham linear algebra appl 214 1995 193 here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b moreover we consider the nearness to rank deficiency finally we give some decomposition tools and their application in the regularized weighted tikhonov problems

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

the lie algebra of homeomorphisms of the circle we define and study an infinite dimensional lie algebrahomeo which is shown to be naturally associated to thetopologicallie grouphomeo of all orientation preserving homeomorphisms of the circle roughly we rely on the universal decorated teichmuller theory developed before as motivation to provide frechet coordinates on the homogeneous space given byhomeo modulo the group of real fractional linear transformations whose corresponding vector fields on the circle we then extend by the usual lie algebrasl2of real traceless two by two matrices in order to definehomeo surprisingly homeo turns out to be equal to the algebra of all vector fields on the circle which are piecewisesl2 in the obvious sense it is evidently important to consider the relationship between our new frechet coordinates and the usual trigonometric functions on the circle and we undertake here both natural infinitesimal calculations we finally apply some further previous work in order to give sufficient conditions on the fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory

volatility of linear and nonlinear time series previous studies indicated that nonlinear properties of gaussian distributed time series with long range correlations u i can be detected and quantified by studying the correlations in the magnitude series ensuremath mid u i ensuremath mid the volatility however the origin for this empirical observation still remains unclear and the exact relation between the correlations in u i and the correlations in ensuremath mid u i ensuremath mid is still unknown here we develop analytical relations between the scaling exponent of linear series u i and its magnitude series ensuremath mid u i ensuremath mid moreover we find that nonlinear time series exhibit stronger or the same correlations in the magnitude time series compared with linear time series with the same two point correlations based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series the nonlinear multifractal time series is generated by multiplying a long range correlated time series that represents the magnitude series with uncorrelated time series that represents the sign series mathrm sgn u i we apply our techniques on daily deep ocean temperature records from the equatorial pacific the region of the el nin o phenomenon and find i long range correlations from several days to several years with 1 f power spectrum ii significant nonlinear behavior as expressed by long range correlations of the volatility series and iii broad multifractal spectrum

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

magnitude and sign of long range correlated time series decomposition and surrogate signal generation we systematically study the scaling properties of the magnitude and sign of the fluctuations in correlated time series which is a simple and useful approach to distinguish between systems with different dynamical properties but the same linear correlations first we decompose artificial long range power law linearly correlated time series into magnitude and sign series derived from the consecutive increments in the original series and we study their correlation properties we find analytical expressions for the correlation exponent of the sign series as a function of the exponent of the original series such expressions are necessary for modeling surrogate time series with desired scaling properties next we study linear and nonlinear correlation properties of series composed as products of independent magnitude and sign series these surrogate series can be considered as a zero order approximation to the analysis of the coupling of magnitude and sign in real data a problem still open in many fields we find analytical results for the scaling behavior of the composed series as a function of the correlation exponents of the magnitude and sign series used in the composition and we determine the ranges of magnitude and sign correlation exponents leading to either single scaling or to crossover behaviors finally we obtain how the linear and nonlinear properties of the composed series depend on the correlation exponents of their magnitude and sign series based on this information we propose a method to generate surrogate series with controlled correlation exponent and multifractal spectrum

a representation of the general common solution to the matrix equations a1xb1 c1 and a2xb2 c2 with applications we give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations a1xb1 c1 and a2xb2 c2 and derive a new representation of the general common solution to these two equations we apply this result to determine new necessary and sufficient conditions for the existence of an hermitian solution and a representation of the general hermitian solution to the matrix equation axb c

infinitesimal bending and twisting in one dimensional dynamics an infinitesimal theory for bending and earthquaking in one dimensional dynamics is developed it is shown that any tangent vector to teichmuller space is the initial data for a bending and for an earthquaking or dinary differential equation the discussion involves an analysis of infinitesimal bendings and earthquakes the hilbert transform natural bounded linear oper ators from a banach space of measures on the mobius strip to tangent vectors to teichmuller space and the construction of a nonlinear right inverse for these operators the inverse is constructed by establishing an infinitesimal version of thurston s earthquake theorem

on hilbert fourier and wavelet transforms several new families of wavelets which are derived from earlier work by the author on teichmuller theory are introduced and studied these wavelets are associated with the modular group rather than the affine group of the real line as for classical wavelets in further contrast to the usual setting there are explicit and essentially algebraic formulas relating wavelet and fourier expansions furthermore rather than relying on equally spaced multiscale sampling methods the wavelet expansions introduced here rely instead on an explicit nonuniformly spaced multiscale sampling method that depends upon the farey sequence of classical number theory these ingredients combine to give novel methods for calculating wavelet fourier and hilbert transforms with potential applications to myriad problems in electrical engineering by employing a new characteristic length associated to 2 2 matrices finiteness and convergence properties of the wavelet expansions are derived for functions with more than 5 2 derivatives in l2 one of our wavelet families forms a frame and the hilbert transform is calculable in hyperbolic coordinates 2002 wiley periodicals inc

effect of extreme data loss on long range correlated and anticorrelated signals quantified by detrended fluctuation analysis detrended fluctuation analysis dfa is an improved method of classical fluctuation analysis for nonstationary signals where embedded polynomial trends mask the intrinsic correlation properties of the fluctuations to better identify the intrinsic correlation properties of real world signals where a large amount of data is missing or removed due to artifacts we investigate how extreme data loss affects the scaling behavior of long range power law correlated and anticorrelated signals we introduce a segmentation approach to generate surrogate signals by randomly removing data segments from stationary signals with different types of long range correlations the surrogate signals we generate are characterized by four parameters i the dfa scaling exponent alpha of the original correlated signal u i ii the percentage p of the data removed from u i iii the average length mu of the removed or remaining data segments and iv the functional form p l of the distribution of the length l of the removed or remaining data segments we find that the global scaling exponent of positively correlated signals remains practically unchanged even for extreme data loss of up to 90 in contrast the global scaling of anticorrelated signals changes to uncorrelated behavior even when a very small fraction of the data is lost these observations are confirmed on two examples of real world signals human gait and commodity price fluctuations we further systematically study the local scaling behavior of surrogate signals with missing data to reveal subtle deviations across scales we find that for anticorrelated signals even 10 of data loss leads to significant monotonic deviations in the local scaling at large scales from the original anticorrelated to uncorrelated behavior in contrast positively correlated signals show no observable changes in the local scaling for up to 65 of data loss while for larger percentage of data loss the local scaling shows overestimated regions with higher local exponent at small scales followed by underestimated regions with lower local exponent at large scales finally we investigate how the scaling is affected by the average length probability distribution and percentage of the remaining data segments in comparison to the removed segments we find that the average length mu r of the remaining segments is the key parameter which determines the scales at which the local scaling exponent has a maximum deviation from its original value interestingly the scales where the maximum deviation occurs follow a power law relationship with mu r whereas the percentage of data loss determines the extent of the deviation the results presented in this paper are useful to correctly interpret the scaling properties obtained from signals with extreme data loss

diff s 1 and the teichmuller spaces precisely two of the homogeneous spaces that appear as coadjoint orbits of the group of string reparametrizations widehat diff s 1 1 carry in a natural way the structure of infinite dimensional holomorphically homogeneous complex analytic kahler manifolds these aren diff s1 rot s1 andm diff s1 mob s1 note thatn is a holomorphic disc fiber space overm now m can be naturally considered as embedded in the classical universal teichmuller spacet 1 simply by noting that a diffeomorphism ofs1 is a quasisymmetric homeomorphism t 1 is itself a homomorphically homogeneous complex banach manifold we prove in the first part of the paper that the inclusion ofm int 1 iscomplex analytic

phase transition in random walks with long range correlations motivated by recent results in the theory of correlated sequences we analyze the dynamics of random walks with long term memory binary chains with long range correlations in our model the probability for a unit bit in a binary string depends on the fraction of unities preceding it we show that the system undergoes a dynamical phase transition from normal diffusion in which the variance d l scales as the string s length l into a superdiffusion phase d l approximately lalpha alpha 1 when the correlation strength exceeds a critical value we demonstrate the generality of our results with respect to alternative models and discuss their applicability to various data such as coarse grained dna sequences written texts and financial data

effects of coarse graining on the scaling behavior of long range correlated and anti correlated signals abstract we investigate how various coarse graining signal quantization methods affect the scaling properties of long range power law correlated and anti correlated signals quantified by the detrended fluctuation analysis specifically for coarse graining in the magnitude of a signal we consider i the floor ii the symmetry and iii the centro symmetry coarse graining methods we find that for anti correlated signals coarse graining in the magnitude leads to a crossover to random behavior at large scales and that with increasing the width of the coarse graining partition interval this crossover moves to intermediate and small scales in contrast the scaling of positively correlated signals is less affected by the coarse graining with no observable changes when 1 while for 1 a crossover appears at small scales and moves to intermediate and large scales with increasing for very rough coarse graining 3 based on the floor and symmetry methods the position of the crossover stabilizes in contrast to the centro symmetry method where the crossover continuously moves across scales and leads to a random behavior at all scales thus indicating a much stronger effect of the centro symmetry compared to the floor and the symmetry method for coarse graining in time where data points are averaged in non overlapping time windows we find that the scaling for both anti correlated and positively correlated signals is practically preserved the results of our simulations are useful for the correct interpretation of the correlation and scaling properties of symbolic sequences

nonlinear strength quantifier based on phase correlation abstract distinguishing linear series from nonlinear ones is of great importance for not only understanding the processes generating these series but also their modeling and prediction there are many methods and their nonlinear quantifiers to reach this goal but no one is powerful enough to deal with any given series in this study the artificially generating series with given tailored phase correlation strength is applied to test the performance of some widely used methods and their nonlinear quantifiers given in the previous studies both advantages and shortcomings are shown for these methods and their nonlinear metrics especially spurious results from these methods and their nonlinear metrics can be found for some cases much weak or strong phase correlations in order to avoid these spurious results modified phase correlation quantifiers are proposed at the same time multiple stripes found in the phase map can be explained from the view point of enhanced nonlinear strength

coadjoint orbits of the virasoro group the coadjoint orbits of the virasoro group which have been investigated by lazutkin and pankratova and by segal should according to the kirillov kostant theory be related to the unitary representations of the virasoro group in this paper the classification of orbits is reconsidered with an explicit description of the possible centralizers of coadjoint orbits the possible orbits are diff s1 itself diff s1 s1 and diff s1 sl n 2 r withsl n 2 r a certain discrete series of embeddings ofsl 2 r in diff s1 and diffs1 t wheret may be any of certain rather special one parameter subgroups of diffs1 an attempt is made to clarify the relation between orbits and representations it appears that quantization of diffs1 s1 is related to unitary representations with nondegenerate kac determinant unitary verma modules while quantization of diffs1 sl n 2 r is seemingly related to unitary representations with null vectors in leveln a better understanding of how to quantize the relevant orbits might lead to a better geometrical understanding of virasoro representation theory in the process of investigating virasoro coadjoint orbits we observe the existence of left invariant symplectic structures on the virasoro group manifold as is described in an appendix these give rise to lie bialgebra structures with the virasoro algebra as the underlying lie algebra

effect of trends on detrended fluctuation analysis scaling behavior we find that crossovers result from the competition between the scaling of the noise and the apparent scaling of the trend we study how the characteristics of these crossovers depend on i the slope of the linear trend ii the amplitude and period of the periodic trend iii the amplitude and power of the power law trend and iv the length as well as the correlation properties of the noise surprisingly we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws i e long range power law dependence of the position of the crossover on the parameters of the trends we show that the dfa result of noise with a trend can be exactly determined by the superposition of the separate results of the dfa on the noise and on the trend assuming that the noise and the trend are not correlated if this superposition rule is not followed this is an indication that the noise and the superposed trend are not independent so that removing the trend could lead to changes in the correlation properties of the noise in addition we show how to use dfa appropriately to minimize the effects of trends how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation or if the crossover is due to a trend without any transition in the dynamical properties of the noise

vector least squares solutions for coupled singular matrix equations the weighted least squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition in this paper a family of algorithms are applied to solve these problems based on the kronecker structures subsequently we construct a computationally efficient solutions of coupled restricted singular matrix equations furthermore the need to compute the weighted drazin and weighted moore penrose inverses and the use of tian s work and lev ari s results are due to appearance in the solutions of these problems the several special cases of these problems are also considered which includes the well known coupled sylvester matrix equations finally we recover the iterative methods to the weighted case in order to obtain the minimum d norm g vector least squares solutions for the coupled sylvester matrix equations and the results lead to the least squares solutions and invertible solutions as a special case

real and complex earthquakes we consider real earthquakes and by their extensions complex earthquakes of the hyperbolic plane h 2 we show that an earthquake restricted to the boundary s 1 of h 2 is a quasisymmetric map if and only if its earthquake measure is bounded multiplying an earthquake measure by a positive parameter we obtain an earthquake path consequently an earthquake path with a bounded measure is a path in the universal teichmiiller space we extend the real parameter for a bounded earthquake into the complex parameter with small imaginary part such obtained complex earthquake or bending is holomorphic in the parameter moreover the restrictions to s 1 of a bending with complex parameter of small imaginary part is a holomorphic motion of s 1 in the complex plane in particular a real earthquake path with bounded earthquake measure is analytic in its parameter

a characterization for the w weighted drazin inverse and a cramer rule for the w weighted drazin inverse solution we establish characterization for the w weighted drazin inverse of an arbitrary rectangular matrix which reduces to the well known result if the matrix is nonsingular also a cramer rule for finding the unique w weighted drazin inverse solution x r aw k 1 of special restricted linear equationswawx b b r wa k 2 is presented and reduces to the classical cramer rule if a is invertible

integral representation of the w weighted drazin inverse two integral representations are presented for the w weighted drazin inverse a d w of rectangular matrices a w the first one has some restriction on the eigenvalues of aw or wa the second one holds without any restriction on the eigenvalues

effect of nonstationarities on detrended fluctuation analysis detrended fluctuation analysis dfa is a scaling analysis method used to quantify long range power law correlations in signals many physical and biological signals are noisy heterogeneous and exhibit different types of nonstationarities which can affect the correlation properties of these signals we systematically study the effects of three types of nonstationarities often encountered in real data specifically we consider nonstationary sequences formed in three ways i stitching together segments of data obtained from discontinuous experimental recordings or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts a cutting procedure commonly used in preparing data prior to signal analysis ii adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes and iii generating a signal comprised of segments with different properties e g different standard deviations or different correlation exponents we compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities we find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on a the fraction and size of the parts cut out from the signal b the concentration of spikes and their amplitudes c the proportion between segments with different standard deviations or different correlations and d the correlation properties of the stationary signal we show how to develop strategies for preprocessing raw data prior to analysis which will minimize the effects of nonstationarities on the scaling properties of the data and how to interpret the results of dfa for complex signals with different local characteristics