the mathematical analysis of the structure of some songs abstract the mathematical structures of six songs were studied by treating them as complex systems the studies were based on i scattering diagram ii spiral structure iii graph theory and animal diagrams and iv entropy and organization in i the fractal dimension and the radius of gyration was determined for each melody in ii it was shown that the total number of notes used in a song display a spiral which can be expressed by a mathematical equation in iii the animal diagram was obtained by using the pathways between the consecutive notes of the song a ratio defined as the number of edges to the total number of unit edge for the animal diagram was used as a parameter to characterize a melody in iv the entropy the maximum entropy and the negentropy or the organization of the songs were determined in addition the rate of change of organization of the songs was also determined

ambiguities in estimates of critical exponents for long range dependent processes abstract in this article we show that the hurst parameter and a well known wavelet estimator fail to distinguish on off models with substantially different distributions of on times we identify the problem as arising when standard power spectrum techniques which are well defined for commonly considered processes are applied to processes with extremely heavy tailed on times we provide an elementary mechanism whereby such extremely heavy tailed on time processes arise in a simple queuing model

multifractal properties of the indian financial market abstract we investigate the multifractal properties of the logarithmic returns of the indian financial indices bse nse by applying the multifractal detrended fluctuation analysis the results are compared with that of the us s p 500 index numerically we find that q th order generalized hurst exponents h q and q change with the moments q the nonlinear dependence of these scaling exponents and the singularity spectrum f show that the returns possess multifractality by comparing the mf dfa results of the original series to those for the shuffled series we find that the multifractality is due to the contributions of long range correlations as well as the broad probability density function the financial markets studied here are compared with the binomial multifractal model bmfm and have a smaller multifractal strength than the bmfm

on subordinacy and spectral multiplicity for a class of singular differential operators the spectral multiplicity of self adjoint operators h associated with singular differential expressions of the form is investigated based on earlier work of i s kac and recent results on subordinacy complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of i the boundary behaviour of titchmarsh weyl m functions and ii the asymptotic properties of solutions of lu u at the endpoints a and b in particular it is shown that h has multiplicity two if and only if l is in the limit point case at both a and b and the set of all for which no solution of lu u is subordinate at either a or b has positive lebesgue measure the results are completely general subject only to minimal restrictions on the coefficients p r q r and w r and the assumption of separated boundary conditions when l is in the limit circle case at both endpoints

right limits and reflectionless measures for cmv matrices we study cmv matrices by focusing on their right limit sets we prove a cmv version of a recent result of remling dealing with the implications of the existence of absolutely continuous spectrum and we study some of its consequences we further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of khrushchev to demonstrate the analogy with the jacobi case we recover corresponding previous results of simon using the same approach

temporal series analysis approach to spectra of complex networks the spacing of nearest levels of the spectrum of a complex network can be regarded as a time series joint use of the multifractal detrended fluctuation approach mf dfa and diffusion entropy de is employed to extract characteristics from this time series for the watts strogatz small world model there exists a critical point at rewiring probability p r 0 32 for a network generated in the range 0l p r l0 32 the correlation exponent is in the range of 1 0 char21 1 64 above this critical point all the networks behave similar to that at p r 1 for the erdos renyi model the time series behaves like fractional brownian motion noise at p text er 1 n for the growing random network grn model the values of the long range correlation exponent are in the range of 0 74 char21 0 83 for most of the grn networks the probability distribution function of a constructed time series obeys a gaussian form in the joint use of mf dfa and de the shuffling procedure in de is essential to obtain a reliable result

reflectionless sturm liouville equations we consider compactly supported perturbations of periodic sturm liouville equations in this context one can use the floquet solutions of the periodic background to define scattering coefficients we prove that if the reflection coefficient is identically zero then the operators corresponding to the periodic and perturbed equations respectively are unitarily equivalent in some appendices we also provide the proofs of several basic estimates e g bounds and asymptotics for the relevant m functions

on isospectral sets of jacobi operators we consider the inverse spectral problem for a class of reflectionless bounded jacobi operators with empty singularly continuous spectra our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum the corresponding isospectral set of jacobi operators is explicitly determined in terms of dirichlet type data

natural boundaries and spectral theory we present and exploit an analogy between lack of absolutely continuous spectrum for schrodinger operators and nat ural boundaries for power series among our new results are gen eralizations of hecke s example and natural boundary examples for random power series where independence is not assumed

characterization of sleep stages by correlations in the magnitude and sign of heartbeat increments we study correlation properties of the magnitude and the sign of the increments in the time intervals between successive heartbeats during light sleep deep sleep and rapid eye movement rem sleep using the detrended fluctuation analysis method we find short range anticorrelations in the sign time series which are strong during deep sleep weaker during light sleep and even weaker during rem sleep in contrast we find long range positive correlations in the magnitude time series which are strong during rem sleep and weaker during light sleep we observe uncorrelated behavior for the magnitude during deep sleep since the magnitude series relates to the nonlinear properties of the original time series while the sign series relates to the linear properties our findings suggest that the nonlinear properties of the heartbeat dynamics are more pronounced during rem sleep thus the sign and the magnitude series provide information which is useful in distinguishing between the sleep stages

long range correlations in the electric signals that precede rupture further investigations the correlations within the time series of the seismic electric signal ses activities have been studied in a previous paper p varotsos n sarlis and e skordas phys rev e 66 011902 2002 here we analyze the time series of successive high and low level states durations the existence of correlation between the states is investigated by means of hurst and detrended fluctuation analysis dfa the multifractal dfa mf dfa is also employed the results point to a stronger correlation and hence longer memory in the series of the high level states furthermore an analysis in the natural time domain reveals that certain power spectrum characteristics seem to distinguish ses activities from artificial man made electric noises more precisely for natural frequencies 0 f 0 5 the curves of the ses activities and artificial noises lie above and below respectively that of the uniform distribution ud a classification of these two types of electric signals ses activities artificial noises cannot be achieved on the basis of the values of the power law exponents alone if the hurst analysis dfa and mf dfa are applied to the original time series the latter two methods however seem to allow a distinction between the ses activities and artificial noises when treating them not in conventional the time frame but in the natural time domain to further test the techniques a time series produced by another system was examined we chose a signal of ion current fluctuations in membrane channels icfmcs the following conclusions among others have been obtained first the power spectrum analysis in the natural time domain shows that the icfmc curve almost coincides in the range 0 f 0 5 with that of the ud and hence icfmc lies just in the boundary between the ses activities and artificial noises second mf dfa indicates monofractality for the icfmcs with a generalized hurst exponent h 50 8460 03 in the range 7 70 ms

spectral properties of a class of reflectionless schrodinger operators abstract we prove that one dimensional reflectionless schrodinger operators with spectrum a homogeneous set in the sense of carleson belonging to the class introduced by sodin and yuditskii have purely absolutely continuous spectra this class includes all earlier examples of reflectionless almost periodic schrodinger operators in addition we construct examples of reflectionless schrodinger operators with more general types of spectra given by the complement of a denjoy widom type domain in c which exhibit a singular component

extraction of backgrounds in fluctuating systems we undertake an extensive analytical study on the generalized detrended fluctuation analysis method designed to detect the scaling behaviors of fluctuating systems but exclude out the influences of the backgrounds or the trends through our extensive studies we systematically extract out the exact backgrounds or the trends of the fluctuating systems to any order expressed in terms of the legendre polynomial our results are exact and can be applied to any 1 1 dimensional continuous fluctuating systems

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

the scaling properties of stock markets based on modified multiscale multifractal detrended fluctuation analysis abstract here we propose the new method dh mma based on multiscale multifractal detrended fluctuation analysis mma to investigate the scaling properties in stock markets it is demonstrated that our approach can provide a more stable and faithful description of the scaling properties in comprehensive range rather than fixing the window length and slide length it allows the assessment of more universal and subtle scaling characteristics we illustrate dh mma by selecting power law artificial data sets and six stock markets from us and china the us stocks exhibit very strong multifractality for positive values of q however the chinese stocks show stronger multifractality for negative q than positive q in general the us stock markets show similar behaviors but chinese stock markets display distinguishing characteristics

complexity entropy causality plane a useful approach for distinguishing songs nowadays we are often faced with huge databases resulting from the rapid growth of data storage technologies this is particularly true when dealing with music databases in this context it is essential to have techniques and tools able to discriminate properties from these massive sets in this work we report on a statistical analysis of more than ten thousand songs aiming to obtain a complexity hierarchy our approach is based on the estimation of the permutation entropy combined with an intensive complexity measure building up the complexity entropy causality plane the results obtained indicate that this representation space is very promising to discriminate songs as well as to allow a relative quantitative comparison among songs additionally we believe that the here reported method may be applied in practical situations since it is simple robust and has a fast numerical implementation

self similarity of the 1 f noise called music abstract suggestions have been made that computer musicians should attempt to compose fractal music and questions have been raised whether there is such a thing as fractal music voss and clark observed that music is scaling or 1 f noise as analyzed on the basis of the amplitude loudness of the audio signals they failed to find a fractal distribution of acoustic frequencies music notes in music analyzing bach s and mozart s compositions we have shown that the incidence of the frequency intervals or of the changes of acoustic frequency has a fractal geometry fractal phenomena are characterized by scale independency the purpose of this investigation is to demonstrate the self similarity of music and to explore its implications

scaling analysis of trends using dfa we study two aspects of the detrended uctuation analysis dfa method namely the scaling behavior of the leading terms of the best 0t polynomials and the detection of trends we show analytically and numerically that the standard deviation of the leading terms of the best 0t polynomials used in dfa displays scaling behavior furthermore we demonstrate that the distribution of these terms can be used to reveal the presence of trends in the data we also argue that this distribution can be used as a sensitive tool for identifying weak trends c 2001 elsevier science b v all rights reserved

kotani last problem and hardy spaces on surfaces of widom type this is a small theory of non almost periodic ergodic families of jacobi matrices with purely however absolutely continuous spectrum the reason why this effect may happen is that under our axioms we found an analytic condition on the resolvent set that is responsible for exactly equivalent to this effect

statistical properties of the yuan exchange rate index abstract we choice the yuan exchange rate index based on a basket of currencies as the effective exchange rate of the yuan and investigate the statistical properties of the yuan exchange rate index after china s exchange rate system reform on the 21st july 2005 after dividing the time series into two parts according to the change in the yuan exchange rate regime in july 2008 we compare the statistical properties of the yuan exchange rate index during these two periods we find that the distribution of the two return series has the exponential form we also perform the detrending moving average analysis dma and the multifractal detrending moving average analysis mfdma the two periods possess different degrees of long range correlations and the multifractal nature is also unveiled in these two time series significant difference is found in the scaling exponents q and singularity spectra f of the two periods obtained from the mfdma analysis besides in order to detect the sources of multifractality shuffling and phase randomization procedures are applied to destroy the long range temporal correlation and fat tailed distribution of the yuan exchange rate index respectively we find that the fat tailedness plays a critical role in the sources of multifractality in the first period while the long memory is the major cause in the second period the results suggest that the change in china s exchange rate regime in july 2008 gives rise to the different multifractal properties of the yuan exchange rate index in these two periods and thus has an effect on the effective exchange rate of the yuan after the exchange rate reform on the 21st july 2005

a note on reflectionless jacobi matrices the property that a jacobi matrix is reflectionless is usually characterized either in terms of weyl m functions or the vanishing of the real part of the boundary values of the diagonal matrix elements of the resolvent we introduce a characterization in terms of stationary scattering theory the vanishing of the reflection coefficients and prove that this characterization is equivalent to the usual ones we also show that the new characterization is equivalent to the notion of being dynamically reflectionless thus providing a short proof of an important result of breuer et al commun math phys 295 531 550 2010 the motivation for the new characterization comes from recent studies of the non equilibrium statistical mechanics of the electronic black box model and we elaborate on this connection

universal scaling behaviors of meteorological variables volatility and relations with original records abstract volatility series defined as the magnitude of the increments between successive elements of five different meteorological variables over china are analyzed by means of detrended fluctuation analysis dfa for short universal scaling behaviors are found in all volatility records whose scaling exponents take similar distributions with similar mean values and standard deviations to reconfirm the relation between long range correlations in volatility and nonlinearity in original series dfa is also applied to the magnitude records defined as the absolute values of the original records the results clearly indicate that the nonlinearity of the original series is more pronounced in the magnitude series

measuring complexity nonextensivity and chaos in the dna sequence of the major histocompatibility complex abstract we analyze 4 mb sequences of the major histocompatibility complex mhc which is a dna segment on chromosome 6 with high gene density controlling many immunological functions and associated with many diseases the analysis is based on modern theoretical and mathematical tools of complexity theory such as nonlinear time series analysis and tsallis non extensive statistics the results revealed that the dna complexity and self organization can be related to fractional dynamical nonlinear processes with low dimensional deterministic chaotic and non extensive statistical character which generate the dna sequences under the extremization of tsallis q entropy principle while it still remains an open question as to whether the dna walk is a fractional brownian motion fbm a static anomalous diffusion process or a non gaussian dynamical fractional anomalous diffusion process the results of this study testify for the latter providing also a possible explanation for the previously observed long range power law correlations of nucleotides as well as the long range correlation properties of coding and non coding sequences present in dna sequences

critical lieb thirring bounds in gaps and the generalized nevai conjecture for finite gap jacobi matrices we prove bounds of the form e i d h dist e e h 1 2 l 1 norm of a perturbation where i is a gap included are gaps in continuum one dimensional periodic schrodinger operators and finite gap jacobi matrices where we get a generalized nevai conjecture about an l 1 condition implying a szeg condition one key is a general new form of the birman schwinger bound in gaps

on a theorem of kac and gilbert we prove a general operator theoretic result that asserts that many multiplicity two selfadjoint operators have simple singular spectrum

effects of non stationarity on the magnitude and sign scaling in the multi scale vertical velocity increment abstract we study the scaling properties of multi scale increment series from the stationary and non stationary vertical velocity records by the detrended fluctuation analysis dfa and series decomposition methods and show the impact of non stationarity on the magnitude and sign scaling properties firstly we show it is difficult to define a significant scaling range for stationary records themselves secondly for increment series with lag one the nonlinear correlation found in magnitude fluctuation i e volatility fluctuation is obvious for both cases and over the smaller scale range the linear correlation found usually in the sign series is the same for both cases however over the larger scale range the difference is sharp and marked nonlinear correlation is also incorporated for the non stationary sign series thirdly with the lags in increment increasing there exist two scaling ranges in both the magnitude and sign fluctuations the nonlinear correlation in the magnitude series over both small and large scale ranges decreases with the lags increasing for the sign series the linear correlation is dominant over both small and large scale ranges for stationary case however the nonlinear correlation can be found over the large scale ranges for non stationary case and will decrease with lags increasing the nonlinear correlation found in the sign series over large scale range is resulted from the modulation of the large scale structures found in the non stationary data which will incorporate parts of the magnitude information into the sign series

detrending fluctuation analysis based on high pass filtering abstract detrended fluctuation analysis dfa is a scaling method commonly used for detecting long range correlations in non stationary time series the dfa method uses a trend based on polynomial fitting to extract and quantify fluctuations at different time scales basically such procedure acts as a non dynamical high pass filter that removes time series components below a given time scale as an alternative to the polynomial fitting approach this paper proposes a dfa method based on well known high pass filters e g butterworth elliptic etc numerical results show that the proposed dfa approach yields results similar to traditional dfa method maybe the main advantage of the proposed dfa method is that efficient implementations of high pass filters are available commercially

bifractality of human dna strand asymmetry profiles results from transcription we use the wavelet transform modulus maxima method to investigate the multifractal properties of strand asymmetry dna walk profiles in the human genome this study reveals the bifractal nature of these profiles which involve two competing scale invariant up to repeat masked distances ensuremath lesssim 40 phantom rule 0 3em 0ex mathrm kbp components characterized by h older exponents h 1 0 78 and h 2 1 respectively the former corresponds to the long range correlated homogeneous fluctuations previously observed in dna walks generated with structural codings the latter is associated with the presence of jumps in the original strand asymmetry noisy signal s we show that a majority of upward downward jumps colocate with gene transcription start end sites here 7228 human gene transcription start sites from the refgene database are found within 2 phantom rule 0 3em 0ex mathrm kbp from an upward jump of amplitude ensuremath delta s ensuremath geqslant 0 1 which suggests that about 36 of annotated human genes present significant transcription induced strand asymmetry and very likely high expression rate

orthogonal polynomials and quadrature rules on the unit circle associated with perturbations of symmetric measures abstract it was shown recently that given a pair of real sequences c n n 1 d n n 1 with d n n 1 a positive chain sequence we can associate a unique nontrivial probability measure on the unit circle and conversely in this paper we consider the set q c of all nontrivial probability measures for which c n 1 n c with c r we show that every measure q c can be obtained from a perturbation of a symmetric measure on 1 1 moreover the sequence of orthogonal polynomials associated with can be given in terms of a perturbation of symmetric orthogonal polynomials associated with we also prove that every measure in q c is quasi symmetric that is there exists a complex valued function q c z satisfying d z q c z d z and such that q c z 1 when c 0 quadrature rules with quasi symmetric weights are also considered finally some examples of orthogonal polynomials on the unit circle and its associated quasi symmetric nontrivial probability measures are obtained

samplesearch importance sampling in presence of determinism the paper focuses on developing effective importance sampling algorithms for mixed probabilistic and deterministic graphical models the use of importance sampling in such graphical models is problematic because it generates many useless zero weight samples which are rejected yielding an inefficient sampling process to address this rejection problem we propose the samplesearch scheme that augments sampling with systematic constraint based backtracking search we characterize the bias introduced by the combination of search with sampling and derive a weighting scheme which yields an unbiased estimate of the desired statistics e g probability of evidence when computing the weights exactly is too complex we propose an approximation which has a weaker guarantee of asymptotic unbiasedness we present results of an extensive empirical evaluation demonstrating that samplesearch outperforms other schemes in presence of significant amount of determinism

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

detrended fluctuation analysis as a regression framework estimating dependence at different scales we propose a framework combining detrended fluctuation analysis with standard regression methodology the method is built on detrended variances and covariances and it is designed to estimate regression parameters at different scales and under potential nonstationarity and power law correlations the former feature allows for distinguishing between effects for a pair of variables from different temporal perspectives the latter ones make the method a significant improvement over the standard least squares estimation theoretical claims are supported by monte carlo simulations the method is then applied on selected examples from physics finance environmental science and epidemiology for most of the studied cases the relationship between variables of interest varies strongly across scales

music walk fractal geometry in music abstract in this study sequences of musical notes from various pieces of music are converted into one variable random walks here termed music walks quantitative measurements of the properties of each musical composition are then performed by applying hurst exponent and fourier spectral analyses on these music walk sequences our results show that music shares the similar fractal properties of a fractional brownian motion fbm that is music displays an anti persistent trend in its tone changes melody over decades of musical notes and music sequence exhibits generally the 1 f type spectrum fractal property with apparently two different values in two different temporal scales

long range correlation properties of coding and noncoding dna sequences genbank analysis an open question in computational molecular biology is whether long range correlations are present in both coding and noncoding dna or only in the latter to answer this question we consider all 33301 coding and all 29453 noncoding eukaryotic sequences each of length larger than 512 base pairs bp in the present release of the genbank to dtermine whether there is any statistically significant distinction in their long range correlation properties standard fast fourier transform fft analysis indicates that coding sequences have practically no correlations in the range from 10 bp to 100 bp spectral exponent beta 0 00 0 04 where the uncertainty is two standard deviations in contrast for noncoding sequences the average value of the spectral exponent beta is positive 0 16 0 05 which unambiguously shows the presence of long range correlations we also separately analyze the 874 coding and the 1157 noncoding sequences that have more than 4096 bp and find a larger region of power law behavior we calculate the probability that these two data sets coding and noncoding were drawn from the same distribution and we find that it is less than 10 10 we obtain independent confirmation of these findings using the method of detrended fluctuation analysis dfa which is designed to treat sequences with statistical heterogeneity such as dna s known mosaic structure patchiness arising from the nonstationarity of nucleotide concentration the near perfect agreement between the two independent analysis methods fft and dfa increases the confidence in the reliability of our conclusion

multifractal detrended cross correlation analysis on gold crude oil and foreign exchange rate time series abstract we apply the recently developed multifractal detrended cross correlation analysis method to investigate the cross correlation behavior and fractal nature between two non stationary time series we analyze the daily return price of gold west texas intermediate and brent crude oil foreign exchange rate data over a period of 18 years the cross correlation has been measured from the hurst scaling exponents and the singularity spectrum quantitatively from the results the existence of multifractal cross correlation between all of these time series is found we also found that the cross correlation between gold and oil prices possess uncorrelated behavior and the remaining bivariate time series possess persistent behavior it was observed for five bivariate series that the cross correlation exponents are less than the calculated average generalized hurst exponents ghe for q 0 and greater than ghe when q 0 and for one bivariate series the cross correlation exponent is greater than ghe for all q values

detrended fluctuation analysis of time series of a firing fusimotor neuron abstract we study the interspike intervals isi time series of the spontaneous fusimotor neuron activity by applying the detrended fluctuation analysis that is a modification of the random walk model analysis thus we have found evidence for the white noise characteristics of the isi time series which means that the fusimotor activity does not possess temporal correlations we conclude that such an activity represents the requisite noisy component for occurrence of the stochastic resonance mechanism in the neural coordination of muscle spindles

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

importance sampling based estimation over and or search spaces for graphical models it is well known that the accuracy of importance sampling can be improved by reducing the variance of its sample mean and therefore variance reduction schemes have been the subject of much research in this paper we introduce a family of variance reduction schemes that generalize the sample mean from the conventional or search space to the and or search space for graphical models the new and or sample means allow trading time and space with variance at one end is the and or sample tree mean which has the same time and space complexity as the conventional or sample tree mean but has smaller variance at other end is the and or sample graph mean which requires more time and space to compute but has the smallest variance theoretically we show that the variance is smaller in the and or space because the and or sample mean is defined over a larger virtual sample size compared with the or sample mean empirically we demonstrate that the and or sample mean is far closer to the true mean than the or sample mean

generalized reflection coefficients i consider general reflection coefficients for arbitrary one dimensional whole line differential or difference operators of order 2 these reflection coefficients are semicontinuous functions of the operator their absolute value can only go down when limits are taken this implies a corresponding semicontinuity result for the absolutely continuous spectrum which applies to a very large class of maps in particular we can consider shift maps thus recovering and generalizing a result of last simon and flows of the toda and kdv hierarchies this is new finally i evaluate an attempt at finding a similar general setup that gives the much stronger conclusion of reflectionless limit operators in more specialized situations

deviations from uniform power law scaling in nonstationary time series a classic problem in physics is the analysis of highly nonstationary time series that typically exhibit long range correlations here we test the hypothesis that the scaling properties of the dynamics of healthy physiological systems are more stable than those of pathological systems by studying beat to beat fluctuations in the human heart rate we develop techniques based on the fano factor and allan factor functions as well as on detrended fluctuation analysis for quantifying deviations from uniform power law scaling in nonstationary time series by analyzing extremely long data sets of up to n 10 5 beats for 11 healthy subjects we find that the fluctuations in the heart rate scale approximately uniformly over several temporal orders of magnitude by contrast we find that in data sets of comparable length for 14 subjects with heart disease the fluctuations grow erratically indicating a loss of scaling stability

generating 1 fβ noise with a low dimensional attractor characteristic its significance for atomic vibrations in proteins and cognitive data we propose an iterative correlator with random seeds as a method to generate series characterized by a low dimensional attractor d 4 65 and scaling properties described by a power law p 1 fa where 0 a 2 the method basically introduces an interaction factor between the terms of the series the higher the interaction factor the stronger the correlation the attractor properties disappear at interaction factors generating scaling exponents lower than about a 0 25 the method proved to be useful for describing the nonlinear properties of the atomic vibrations in the main chain of proteins as well as series of data produced in cognitive experiments

price volume multifractal analysis of the moroccan stock market abstract in this paper we analyzed price volume multifractal cross correlations of moroccan stock exchange we chose the period from january 1st 2000 to january 20th 2017 to investigate the multifractal behavior of price change and volume change series then we used multifractal detrended cross correlations analysis method mf dcca and multifractal detrended fluctuation analysis mf dfa to analyze the series we computed bivariate generalized hurst exponent renyi exponent and spectrum of singularity for each pair of indices to measure quantitatively cross correlations furthermore we used detrended cross correlations coefficient dcca and cross correlation test q m to analyze cross correlation quantitatively and qualitatively by analyzing results we found existence of price volume multifractal cross correlations the spectrum width has a strong multifractal cross correlation we remarked that volume change series is anti persistent when we analyzed the generalized hurst exponent for all moments q the cross correlation test showed the presence of a significant cross correlation however dcca coefficient had a small positive value which means that the level of correlation is not very significant finally we analyzed sources of multifractality and their degree of contribution in the series

detrended fluctuation analysis for fractals and multifractals in higher dimensions one dimensional detrended fluctuation analysis dfa and multifractal detrended fluctuation analysis mfdfa are widely used in the scaling analysis of fractal and multifractal time series because they are accurate and easy to implement in this paper we generalize the one dimensional dfa and mfdfa to higher dimensional versions the generalization works well when tested with synthetic surfaces including fractional brownian surfaces and multifractal surfaces the two dimensional mfdfa is also adopted to analyze two images from nature and experiment and nice scaling laws are unraveled

nonlinear properties of the atomic vibrations in protein backbones the temperature factors series of protein backbone atoms resulting from the x ray diffraction analysis of crystal data were analyzed by fast fourier transform detrended fluctuation analysis dfa and correlation dimension the spectral analysis of 50 randomly selected proteins revealed the presence of a long range correlation characterized on average by a power law p 1 f1 79 the dfa showed evidence for long range correlation extending over an average 48 atoms length equivalent to 16 amino acid residues the mean value of the scaling exponent was about 1 35 beyond this range the scaling exponent decreased significantly the correlation dimension analysis revealed the presence of a strange attractor having an average dimension of 4 34 a similar analysis of the temperature factor for proteins in solution as resulting from the nmr analysis revealed that the spectral scaling exponent was lower in solution than in crystal form this was expected from our model described in coza and morariu generating 1 f noise with a low dimensional attractor characteristic its significance for atomic vibration in proteins and cognitive data physica a the attractor dimension was lower by about one unit for proteins in solution compared to the crystal form therefore the nature of at least one of the independent variables controlling the vibrations is dependent on the form of aggregation higher values of the scaling exponents were associated to the inactive or tense state of the protein lower values of the scaling exponent were associated to higher enzyme activity regardless of higher or lower flexibility while the molecular biologist s view stresses the relation of particular sites in the protein to its function the present analysis underlines the unity of the protein molecule

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

reflectionless herglotz functions and jacobi matrices we study several related aspects of reflectionless jacobi matrices first we discuss the singular part of the corresponding spectral measures we then show how to identify sets on which measures are reflectionless by looking at the logarithmic potentials of these measures

equality of the spectral and dynamical definitions of reflection for full line jacobi matrices schrodinger operators and cmv matrices we show that being reflectionless in the sense of the well known property of m functions is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to x as t goes entirely to x as t this allows us to settle a conjecture of deift and simon from 1983 regarding ergodic jacobi matrices

multifractal analyses of music sequences multifractal analysis is applied to study the fractal property of music in this paper a method is proposed to transform both the melody and rhythm of a music piece into individual sets of distributed points along a one dimensional line the structure of the musical composition is thus manifested and characterized by the local clustering pattern of these sequences of points specifically the local holder exponent and the multifractal spectrum are calculated for the transformed music sequences according to the multifractal formalism the observed fluctuations of the holder exponent along the music sequences confirm the non uniformity feature in the structures of melodic and rhythmic motions of music our present result suggests that the shape and opening width of the multifractal spectrum plot can be used to distinguish different styles of music in addition a characteristic curve is constructed by mapping the point sequences converted from the melody and rhythm of a musical work into a two dimensional graph each different pieces of music has its own unique characteristic curve this characteristic curve which also exhibits a fractal trait unveils the intrinsic structure of music

revealing competitive behaviours in music by means of the multifractal detrended fluctuation analysis application to bach s sinfonias the one two and three dimensional multifractal detrended fluctuation analysis mfdfa was applied to bach s sinfonias which are characterized by the superposition of three different voices each voice represented as a time series can be considered as a component of a one two or three dimensional vector the one dimensional mf dfa was applied to any single voice while the two and three dimensional mf dfa was applied to the couples of voices and to the triple respectively each voice is characterized by a multifractal degree md indicated by the range of the generalized hurst exponents the higher the md the larger the amount of heterogeneity and irregularity competitive scaling multifractal behaviours in bach s sinfonias were revealed although one or two voices showed a relatively high md the other two voices or voice are characterized by a low md nevertheless the overall effect of the sinfonia measured by the md of the triple tends towards homogeneity or at least to an average between the different competitive scaling behaviour shown by the different voices

superroughening by linear growth equations with spatiotemporally correlated noise we give an extensive study on a class of interfacial superroughening processes with finite lateral system size in 1 1 dimensions described by linear growth equations with spatiotemporally power law decaying correlated noise since some of these processes have an extremely long relaxation time we first develop a very efficient method capable of simulating the interface morphology of these growth processes even in very late time we numerically observe that this class of superrough growth processes indeed gradually develops macroscopic structures with the lateral size comparable to the lateral system size through the rigorous analytical study of the equal time height difference correlation function the different time height difference correlation function and the local width we explicitly evaluate not only the leading anomalous dynamic scaling term but also all the subleading anomalous dynamic scaling terms which dominate over the ordinary dynamic scaling term moreover the relation between the macroscopic structure formation and anomalous interfacial roughening of the superrough growth processes is analytically investigated in detail

sieved para orthogonal polynomials on the unit circle abstract sieved orthogonal polynomials on the unit circle were introduced independently by ismail and li 1992 15 and marcellan and sansigre 1991 19 we look at the para orthogonal polynomials chain sequences and quadrature formulas that follow from the kernel polynomials of sieved orthogonal polynomials on the unit circle

fractal geometry of music abstract music critics have compared bach s music to the precision of mathematics what mathematics and what precision are the questions for a curious scientist the purpose of this short note is to suggest that the mathematics is at least in part mandelbrot s fractal geometry and the precision is the deviation from a log log linear plot

measure representation and multifractal analysis of complete genomes this paper introduces the notion of measure representation of dna sequences spectral analysis and multifractal analysis are then performed on the measure representations of a large number of complete genomes the main aim of this paper is to discuss the multifractal property of the measure representation and the classification of bacteria from the measure representations and the values of the dq spectra and related cq curves it is concluded that these complete genomes are not random sequences in fact spectral analyses performed indicate that these measure representations considered as time series exhibit strong long range correlation here the long range correlation is for the k strings with dictionary ordering and it is different from the base pair correlations introduced by other people for substrings with length k58 the dq spectra of all organisms studied are multifractal like and sufficiently smooth for the cq curves to be meaningful with the decreasing value of k the multifractality lessens the cq curves of all bacteria resemble a classical phase transition at a critical point but the analogous phase transitions of chromosomes of nonbacteria organisms are different apart from chromosome 1 of c elegans they exhibit the shape of double peaked specific heat function a classification of genomes of bacteria by assigning to each sequence a point in two dimensional space d21 d1 and in three dimensional space d21 d1 d22 was given bacteria that are close phylogenetically are almost close in the spaces d21 d1 and d21 d1 d22

scaling multifractality and long range correlations in well log data of large scale porous media abstract three distinct methods namely the spectral density the multifractal random walk approach and the multifractal detrended fluctuation analysis are utilized to study the properties of four distinct types of well logs from three oil and gas fields namely the natural gamma ray emission neutron porosity bulk density and the sonic transient time logs such well logs have never been analyzed by the methods that we utilize in the present study the results indicate that the well logs exhibit multifractal characteristics and the estimated hurst exponents by the three methods are close to each other using multifractal detrended fluctuation analysis and the shuffled and surrogated data we find that the source of multifractality is due to both broad probability density functions of the data and long range correlations in them the correlations are persistent and are characterized by a hurst exponent h 0 5 despite very significant differences in the geology of the three reservoirs ranging from shaly sands to fractured carbonate reservoirs there is a rough universality in the log data in that the hurst exponents for all the logs vary in a very narrow range

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

treadmill belt width but not feedback from the lower visual field influences the noise characteristics of step width time series abstract step kinematic variability which has been associated with gait related fall risk is thought to be attributed to neuromotor noise altered neuromotor control of step kinematics would be expected to manifest as changes in the noise related characteristics of the step kinematic time series this study determined the effects of eliminating feedback from the lower visual field and reducing treadmill belt width on the noise characteristics of step width time series and statistical measures of step width variability during treadmill walking we hypothesized that eliminating feedback from the lower visual field and reducing treadmill belt width would both alter the noise characteristics of step width time series reflected by decreased fractal scaling and increase statistical measures of step width variability eighteen young adults performed four randomly ordered walking trials during which we manipulated visual feedback from the lower visual field normal and obstructed and treadmill belt width wide and narrow reducing treadmill belt width but not eliminating feedback from the lower visual field significantly reduced the fractal scaling of step width time series indicating a shift towards white uncorrelated noise these results suggest that accounting for the influence of treadmill belt width on step width time series may be an important consideration in both laboratory and clinical settings further work is needed to clarify the effects of vision on measures of step width identify the mechanism s underlying the observed shift towards white uncorrelated noise associated with reduced treadmill belt width and to assess the potential relationship between the noise characteristics of step width time series and fall risk

long correlations and truncated levy walks applied to the study latin american market indices abstract this work is devoted to the study of long correlations and other statistical properties of latin american market indices we concluded that the behavior of the return is compatible with a slow convergence to a gaussian distribution we also detected long range correlations in the absolute value of the return analyzing the effects of working with short data series this fact has relevant consequences in the volatility dynamics

detecting long range correlations in time series of neuronal discharges abstract we have studied the discharge dynamics of dorsal horn neurons dhn by applying the detrended fluctuation analysis dfa and the wavelet transform wt technique we have adopted that discharge dynamics is manifested by the random time series of the interspike intervals isi that is by intervals between two consecutive neuronal electrical activities in all cases studied we found two different power law type behaviors across isi enumeration scale that are separated by a crossover region our results reveal that complex neuronal dynamics may change in the presence of external stimulation which is manifested by changing the noise characteristics that appear before the crossover region the noise after the crossover region is of the 1 f type

mass exponent spectrum analysis of human ecg signals and its application to complexity detection abstract the complexity of electrocardiogram ecg signal may reflect the physiological function and healthy status of the heart in this paper we introduced two novel intermediate parameters of multifractality the mass exponent spectrum curvature and area to characterize the nonlinear complexity of ecg signal these indicators express the nonlinear superposition of the discrepancies of singularity strengths from all the adjacent points of the spectrum curve and thus overall subsets of original fractal structure the evaluation of binomial multifractal sets validated these two variables were entirely effective in exploring the complexity of this time series we then studied the ecg mass exponent spectra taken from the cohorts of healthy ischemia and myocardial infarction mi sufferer based on a large sets of 12 leads recordings and took the statistical averages among each crowd experimental results suggest the two values from healthy ecg are apparently larger than those from the heart diseased while the values from ecg of mi sufferer are much smaller than those from the other two groups as for the ischemia sufferer they are almost of moderate magnitude afterward we compared these new indicators with the nonlinear parameters of singularity spectrum the classification indexes and results of total separating ratios tsr defined in the paper both indicated that our method could achieve a better effect these conclusions may be of some values in early diagnoses and clinical applications

local spectral properties of reflectionless jacobi cmv and schrodinger operators we prove that jacobi cmv and schrodinger operators which are reflectionless on a homogeneous set e in the sense of carleson under the assumption of a blaschke type condition on their discrete spectra accumulating at e have purely absolutely continuous spectrum on e

1 f noise structures in pollocks s drip paintings abstract the fractality and scaling properties of luminance patterns in pollock s drip paintings are studied in this paper the painting is represented as a matrix array where each entry represents the luminance at the corresponding canva position detrended fluctuation analysis dfa is used to characterize fractality and the results show that the whole luminance pattern is indeed fractal interestingly the fractality is associated to 1 f noise structures which can be inducing the aesthetic properties of the artistic object in fact for spatial scales below 30 cm pollock s drip painting are organized as 1 f noise structures which presumably are attuned for pleasant i e aesthetic perception

scaling features of texts images and time series in the given paper we consider the scaling features of long letter sequences like human writings discretized images and discretized 3nancial data using several approaches we show that the symbolic strings and time series being analyzed have a complex multiscale structure and demonstrate di5erent scalings for large and small 6uctuations we discuss complex phenomena in the scaling behavior of partition functions in the case of high frequency dax future data c 2001 elsevier science b v all rights reserved

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

samplesearch a scheme that searches for consistent samples sampling from belief networks which have a substantial number of zero probabilities is problematic mcmc algorithms like gibbs sampling do not converge and importance sampling schemes generate many zero weight samples that are rejected yielding an inefficient sampling process the rejection problem in this paper we propose to augment importance sampling with systematic constraint satisfaction search in order to overcome the rejection problem the resulting samplesearch scheme can be made unbiased by using a computationally expensive weighting scheme to overcome this an approximation is proposed such that the resulting estimator is asymptotically unbiased our empirical results demonstrate the potential of our new scheme

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

a difference equation arising from the trigonometric moment problem having random reflection coefficients an operator theoretic approach abstract operators associated with polynomials orthogonal on the unit circle and their doubly infinite analogs are considered we assume that the elements in the operators are obtained from a stationary stochastic process and we investigate the properties of the spectrum of these operators lyapunov exponents are introduced and we examine their relationship with the absolutely continuous components of the spectrum of the above operators

universal distributions for growth processes in 1 1 dimensions and random matrices we develop a scaling theory for kardar parisi zhang growth in one dimension by a detailed study of the polynuclear growth model in particular we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape these distribution functions are computed using the partition function of gaussian random matrices in a cosine potential

uniqueness theorems in inverse spectral theory for one dimensional schrodinger operators new unique characterization results for the potential v x in connection with schrodinger operators on r and on the half line 0 are proven in terms of appropriate krein spectral shift functions particular results obtained include a generalization of a well known uniqueness theorem of borg and marchenko for schrodinger operators on the half line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for schrodinger operators with confining potentials on the entire real line

scattering for general type dirac systems on the semi axis reflection coefficients and weyl functions abstract we show that for general type self adjoint and skew self adjoint dirac systems on the semi axis weyl functions are unique analytic extensions of the reflection coefficients new results on the extension of the weyl functions to the real axis and on the existence in the skew self adjoint case of the weyl functions follow important procedures to recover general type dirac systems from the weyl functions are applied to the recovery of dirac systems from the reflection coefficients we explicitly recover dirac systems from the rational reflection coefficients as well

detrending fluctuation analysis based on moving average filtering abstract detrended fluctuation analysis dfa is a scaling method commonly used for detecting long range correlations in nonstationary time series applications range from financial time series to physiological data however as the removal of trends in dfa is based on discontinuous polynomial fitting oscillations in the fluctuation function and significant errors in crossover locations can be introduced to reduce the problems induced by discontinuous fitting moving average ma methods have been proposed previously by alesio et al eur j phys b 27 2002 197 in this work a variant of such ma methods is studied specifically the performance and characteristics of a ma method based on central differences is studied some important properties of this ma method are analyzed and illustrated with several artificial and real time series

correlations in economic time series a financial index of the new york stock exchange the sp here f is called the crossover frequency iii we confirm these results using the dfa detrended fluctuation analysis method see fig 2b which allows accurate estimates of exponents independent of local trends

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous