multifractal diffusion entropy analysis on stock volatility in financial markets abstract this paper introduces a generalized diffusion entropy analysis method to analyze long range correlation then applies this method to stock volatility series the method uses the techniques of the diffusion process and renyi entropy to focus on the scaling behaviors of regular volatility and extreme volatility respectively in developed and emerging markets it successfully distinguishes their differences where regular volatility exhibits long range persistence while extreme volatility reveals anti persistence

third neighbor correlators of a one dimensional spin 1 2 heisenberg antiferromagnet we exactly evaluate the third neighbor correlator s j z s j 3 z and all the possible nonzero correlators s j ensuremath alpha s j 1 ensuremath beta s j 2 ensuremath gamma s j 3 ensuremath delta of the one dimensional spin frac 1 2 heisenberg mathrm xxx antiferromagnet in the ground state without magnetic field all the correlators are expressed in terms of certain combinations of logarithm mathrm ln 2 the riemann zeta function ensuremath zeta 3 ensuremath zeta 5 with rational coefficients the results accurately coincide with the numerical ones obtained by the density matrix renormalization group method and the numerical diagonalization

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

comment on multifractal diffusion entropy analysis on stock volatility in financial markets physica a 391 2012 5739 5745 abstract in their recent article multifractal diffusion entropy analysis on stock volatility in financial markets huang shang and zhao 2012 6 suggested a generalization of the diffusion entropy analysis method with the main goal of being able to reveal scaling exponents for multifractal times series the main idea seems to be replacing the shannon entropy by the renyi entropy which is a one parametric family of entropies the authors claim that based on their method they are able to separate long range and short correlations of financial market multifractal time series in this comment i show that the suggested new method does not bring much valuable information in obtaining the correct scaling for a multifractal mono fractal process beyond the original diffusion entropy analysis method i also argue that the mathematical properties of the multifractal diffusion entropy analysis should be carefully explored to avoid possible numerical artefacts when implementing the method in analysis of real sequences of data

string correlation functions of the spin 1 2 heisenberg xxz chain we calculate certain string correlation functions originally introduced as order parameters in integer spin chains for the spin 1 2 xxz heisenberg chain at zero temperature and in the thermodynamic limit for small distances we obtain exact results from bethe ansatz and exact diagonalization whereas in the large distance limit field theoretical arguments yield an asymptotic algebraic decay we also make contact with two point spin correlation functions in the asymptotic limit

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

correlation functions for the heisenberg xxz antiferromagnet the general method for calculation of correlation functions in integrable quantum models has been given in papers 1 2 the correlation function of the third components of local spins for the heisenberg one dimensionalxxz antiferromagnet is calculated in this paper the answer is a series which gives in particular an improved version of the usual perturbative expansion in the anisotropy parameter the remarkable property of the series obtained is that the long distance asymptotics of the correlator is given already by the first term the arguments are given in favour of the convergence of the series

quantum spin chain toeplitz determinants and the fisher hartwig conjecture we consider the one dimensional quantum spin chain which is called the xx model xx0 model or isotropic xy model in a transverse magnetic field we are mainly interested in the entropy of a block of lneighboring spins at zero temperature and of an infinite system we represent the entropy in terms of a toeplitz determinant and calculate the asymptotic analytically we derive the first two terms of the asymptotic decomposition interestingly these two terms of decomposition clearly show a length scale related to the field h

reduced qkz equation and correlation functions of the xxz model correlation functions of the xxz model in the massive and massless regimes are known to satisfy a system of linear equations the main relations among them are the difference equations obtained from the qkz equation by specializing the variables 1 2n as 1 n n 1 1 1 we call it the reduced qkz equation in this article we construct a special family of solutions to this system they can be written as linear combinations of products of two transcendental functions open image in new window with coefficients being rational functions we show that correlation functions of the xxz model in the massive regime are given by these formulas with an appropriate choice of open image in new window we also present a conjectural formula in the massless regime

quantum spin chains and riemann zeta function with odd arguments the riemann zeta function is an important object of number theory we argue that it is related to the heisenberg spin 1 2 anti ferromagnet in the xxx spin chain we study the probability of formation of a ferromagnetic string in the anti ferromagnetic ground state in the thermodynamics limit we prove that for short strings the probability can be expressed in terms of the riemann zeta function with odd arguments

multifractal diffusion entropy analysis optimal bin width of probability histograms abstract in the framework of multifractal diffusion entropy analysis we propose a method for choosing an optimal bin width in histograms generated from underlying probability distributions of interest the method presented uses techniques of renyi s entropy and the mean squared error analysis to discuss the conditions under which the error in the multifractal spectrum estimation is minimal we illustrate the utility of our approach by focusing on a scaling behavior of financial time series in particular we analyze the s p500 stock index as sampled at a daily rate in the time period 1950 2013 in order to demonstrate a strength of the method proposed we compare the multifractal spectrum for various bin widths and show the robustness of the method especially for large values of q for such values other methods in use e g those based on moment estimation tend to fail for heavy tailed data or data with long correlations connection between the spectrum and renyi s q parameter is also discussed and elucidated on a simple example of multiscale time series

exact correlation amplitude for the heisenberg antiferromagnetic chain the exact amplitude for the asymptotic correlation function in the heisenberg antiferromagnetic chain is determined the behaviour of the correlation functions for small xxz anisotropy and the form of finite size corrections to the correlation function are also analysed

multicritical scaling in the magnetic hard square lattice gas the scaling behaviour of the magnetic hard square lattice gas is examined in the vicinity of the multicritical line located by the recent exact solution a simple scaling form is proposed for the singular part of the free energy and corresponding scaling forms derived for the sublattice density difference the magnetisation and the densities these are all shown to be consistent with the exact results when restricted to the exact solution manifold 1 1 the case of the densities this is achieved by proving appropriate elliptic function identities

determinant representation for correlation functions of spin 1 2 xxx and xxz heisenberg magnets we consider zero temperature correlation functions of the spin 1 2 xxz heisenberg chain in the critical regime 1 1 in a magnetic field starting from the algebraic bethe ansatz we derive representations for various correlation functions in term of determinats of fredholm integral operators

quantum kz equation with q 1 and correlation functions of the xxz model in the gapless regime an integral solution to the quantum knizhnik zamolodchikov qkz equation with q 1 is presented upon specialization it leads to a conjectural formula for correlation functions of the xxz model in the gapless regime the validity of this conjecture is verified in special cases including the nearest neighbour correlator with an arbitrary coupling constant and general correlators in the xxx and xy limits

third neighbour and other four point correlation functions of spin 1 2 xxz chain the correlation functions of the spin 1 2 xxz chain in the ground state were expressed in the form of multiple integrals for 1 1 and 1 in particular adjacent four point correlation functions were given as certain four dimensional integrals we show that these integrals can be reduced to polynomials with respect to specific one dimensional integrals the results give the polynomial representation of the third neighbour correlation functions

the quantum inverse scattering method approach to correlation functions the inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with ther matrix of xxx type these models include the one dimensional bose gas and the heisenberg xxx model the algebraic questions of the problem are considered

scaling detection in time series diffusion entropy analysis the methods currently used to determine the scaling exponent of a complex dynamic process described by a time series are based on the numerical evaluation of variance this means that all of them can be safely applied only to the case where ordinary statistical properties hold true even if strange kinetics are involved we illustrate a method of statistical analysis based on the shannon entropy of the diffusion process generated by the time series called diffusion entropy analysis dea we adopt artificial gauss and levy time series as prototypes of ordinary and anomalous statistics respectively and we analyze them with the dea and four ordinary methods of analysis some of which are very popular we show that the dea determines the correct scaling exponent even when the statistical properties as well as the dynamic properties are anomalous the other four methods produce correct results in the gauss case but fail to detect the correct scaling in the case of levy statistics

levy scaling the diffusion entropy analysis applied to dna sequences we address the problem of the statistical analysis of a time series generated by complex dynamics with the diffusion entropy analysis dea n scafetta p hamilton and p grigolini fractals 9 193 2001 this method is based on the evaluation of the shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations rather than on the measurement of the variance of this diffusion process as done with the traditional methods we compare the dea to the traditional methods of scaling detection and prove that the dea is the only method that always yields the correct scaling value if the scaling condition applies furthermore dea detects the real scaling of a time series without requiring any form of detrending we show that the joint use of dea and variance method allows to assess whether a time series is characterized by levy or gauss statistics we apply the dea to the study of dna sequences and prove that their large time scales are characterized by levy statistics regardless of whether they are coding or noncoding sequences we show that the dea is a reliable technique and at the same time we use it to confirm the validity of the dynamic approach to the dna sequences proposed in earlier work

the fisher hartwig conjecture and generalizations abstract we discuss the status of the fisher hartwig conjecture concerning the asymptotic expansion of a class of toeplitz determinants with singular generating functions a counterexample is given for a nonrational generating function and we formulate a generalized fisher hartwig conjecture

asymptotics of toeplitz determinants and the emptiness formation probability for the xy spin chain we study an asymptotic behaviour of a special correlator known as the emptiness formation probability efp for the one dimensional anisotropic xy spin 1 2 chain in a transverse magnetic field this correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models for the xy spin chain the correlator can be expressed as the determinant of a toeplitz matrix and its asymptotical behaviours for n throughout the phase diagram are obtained using known theorems and conjectures on toeplitz determinants we find that the decay is exponential everywhere in the phase diagram of the xy model except on the critical lines i e where the spectrum is gapless in these cases a power law prefactor with a universal exponent arises in addition to an exponential or gaussian decay the latter gaussian behaviour holds on the critical line corresponding to the isotropic xy model while at the critical value of the magnetic field the efp decays exponentially at small anisotropy one has a crossover from the gaussian to the exponential behaviour we study this crossover using the bosonization approach