surrogate testing of linear feedback processes with non gaussian innovations abstract surrogate testing is used widely to determine the nature of the process generating the given empirical sample in the present study the usefulness of phase randomized surrogates amplitude adjusted fourier transform and iterated amplitude adjusted fourier transform surrogates on statistical inference of linearly correlated noise with non gaussian innovations and their static invertible nonlinear transforms from their empirical samples are discussed existing surrogate testing procedures which retain the auto correlation function in the surrogates may not be appropriate in the presence of non gaussian innovations

circle and divisor problems and double series of bessel functions abstract in approximately 1915 ramanujan recorded two identities involving doubly infinite series of bessel functions the identities were brought to the mathematical public for the first time when his lost notebook was published in 1988 and are connected with the classical long standing circle and divisor problems respectively we provide a proof of the first identity for the first time by analytically continuing a new kind of dirichlet series delicate estimates of exponential sums are needed and the new methods we introduce may be of independent interest

a q operator identity for the correlation functions of the infinite xxz spin chain an explicit construction for q operators of the finite xxz spin chain with twisted boundary conditions is presented the massless and the massive regimes are considered as well as the root of unity case it is explained how these results yield an alternative expression for the trace function employed in the description of the correlation functions of the inhomogeneous xxz model on the infinite lattice by boos jimbo miwa smirnov and takeyama

traces on the sklyanin algebra and correlation functions of the eight vertex model we propose a conjectural formula for correlation functions of the z invariant inhomogeneous eight vertex model we refer to this conjecture as ansatz it states that correlation functions are linear combinations of products of three transcendental functions with theta functions and derivatives as coefficients the transcendental functions are essentially logarithmic derivatives of the partition function per site the coefficients are given in terms of a linear functional tr on the sklyanin algebra which interpolates the usual trace on finite dimensional representations we establish the existence of tr and discuss the connection to the geometry of the classical limit we also conjecture that the ansatz satisfies the reduced qkz equation as a non trivial example of the ansatz we present a new formula for the next nearest neighbour correlation functions

on the physical part of the factorized correlation functions of the xxz chain it was recently shown by jimbo et al 2008 arxiv 0811 0439 that the correlation functions of a generalized xxz chain associated with an inhomogeneous six vertex model with a disorder parameter and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions and here we approach the description of the same correlation functions and in particular of the function from a different direction we start from a novel multiple integral representation for the density matrix of a finite chain segment of length m in the presence of a disorder field we explicitly factorize the integrals for m 2 based on this we present an alternative description of the function in terms of the solutions of certain linear and nonlinear integral equations we then prove directly that the two definitions of describe the same function the definition in the work of jimbo et al 2008 arxiv 0811 0439 was crucial for the proof of the factorization the definition given here together with the known description of in terms of the solutions of nonlinear integral equations is useful for performing e g the trotter limit in the finite temperature case or for obtaining numerical results for the correlation functions at short distances we also address the issue of the construction of an exponential form of the density matrix for finite

density matrix elements and entanglement entropy for the spin 1 2 xxz chain at δ 1 2 we have analytically obtained all the density matrix elements up to six lattice sites for the spin 1 2 heisenberg xxz chain at 1 2 we use the multiple integral formula of the correlation function for the massless xxz chain derived by jimbo and miwa as for the spin spin correlation functions we have newly obtained the fourth and fifth neighbour transverse correlation functions we have calculated all the eigenvalues of the density matrix and analyse the eigenvalue distribution using these results the exact values of the entanglement entropy for the reduced density matrix up to six lattice sites have been obtained we observe that our exact results agree quite well with the asymptotic formula predicted by the conformal field theory

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

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

factorization of the ising model form factors we present a general method for analytically factorizing the n fold form factor integrals f n n n t for the correlation functions of the ising model on the diagonal in terms of the hypergeometric functions 2f1 1 2 n 1 2 n 1 t which appear in the form factor f 1 n n t new quadratic recursion and quartic identities are obtained for the form factors for n 2 3 for n 2 3 4 explicit results are given for the form factors these factorizations are proved for all n for n 2 3 these results yield the emergence of palindromic polynomials canonically associated with elliptic curves as a consequence understanding the form factors amounts to describing and understanding an infinite set of palindromic polynomials canonically associated with elliptic curves from an analytical viewpoint the relation of these palindromic polynomials with hypergeometric functions associated with elliptic curves is made very explicitly and from a differential algebra viewpoint this corresponds to the emergence of direct sums of differential operators homomorphic to symmetric powers of a second order operator associated with elliptic curve

a q operator for the twisted xxx model taking the isotropic limit 1 in a recent representation theoretic construction of baxter s q operators for the xxz model with quasi periodic boundary conditions we obtain new results for the xxx model we show that quasi periodic boundary conditions are needed to ensure convergence of the q operator construction and derive a quantum wronskian relation which implies two different sets of bethe ansatz equations one above the other below the equator of total spin sz 0 we discuss the limit to periodic boundary conditions at the end and explain how this construction relates to the trace functional introduced by boos et al in the context of correlation functions on the infinite lattice we also identify a special subclass of solutions to the quantum wronskian and numerically verify them up to spin chains of ten sites this special type of solutions might persist for longer chains

modeling heart rate variability in healthy humans a turbulence analogy many complex systems share similar statistical characteristics in this letter a turbulence analogy is proposed for the long term heart rate variability of healthy humans based on such an analogy the equivalence of an inertial range is found and a cascade model which captures the statistical properties of the heart rate data is given

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

new pathways and connections in number theory and analysis motivated by two incorrect claims of ramanujan abstract the focus of this paper commences with an examination of three not obviously related pages in ramanujan s lost notebook pages 336 335 and 332 in decreasing order of attention on page 336 ramanujan proposes two identities but the formulas are wrong each is vitiated by divergent series we concentrate on only one of the two incorrect identities which may have been devised to attack the extended divisor problem we prove here a corrected version of ramanujan s claim which contains the convergent series appearing in it the convergent series in ramanujan s faulty claim is similar to one used by g f voronoi g h hardy and others in their study of the classical dirichlet divisor problem this now brings us to page 335 which comprises two formulas featuring doubly infinite series of bessel functions the first being conjoined with the classical circle problem initiated by gauss and the second being associated with the dirichlet divisor problem the first and fourth authors along with sun kim have written several papers providing proofs of these two difficult formulas in different interpretations in this monograph we return to these two formulas and examine them in more general settings the aforementioned convergent series in ramanujan s identity is also similar to one that appears in a curious identity found in chapter 15 in ramanujan s second notebook written in a more elegant equivalent formulation on page 332 in the lost notebook this formula may be regarded as a formula for 1 2 and in 1925 s wigert obtained a generalization giving a formula for 1 k for any even integer k 2 we extend the work of ramanujan and wigert in this paper the voronoi summation formula appears prominently in our study in particular we generalize work of j r wilton and derive an analogue involving the sum of divisors function s n the modified bessel functions k s x arise in several contexts as do lommel functions we establish here new series and integral identities involving modified bessel functions and modified lommel functions among other results we establish a modular transformation for an infinite series involving s n and modified lommel functions we also discuss certain obscure related work of n s koshliakov we define and discuss two new related classes of integral transforms which we call koshliakov transforms because he first found elegant special cases of each

weighted divisor sums and bessel function series on page 335 in his lost notebook ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary bessel functions we provide the first published proof of this result the identity yields as corollaries representations of weighted divisor sums in particular the summatory function for r 2 n the number of representations of the positive integer n as a sum of two squares

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

finite temperature correlation functions from discrete functional equations we present a new approach to the static finite temperature correlation functions of the heisenberg chain based on functional equations an inhomogeneous generalization of the n site density operator is considered the lattice path integral formulation with a finite but arbitrary trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters we show that these equations yield a unique characterization of the density operator our functional equations are a discrete version of the reduced q knizhnik zamolodchikov equations which played a central role in the study of the zero temperature case as a natural result and independent of the arguments given by jimbo et al 2009 j phys a math theor 42 304018 we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature they are a sum of products of nearest neighbour correlators

surrogate analysis of volatility series from long range correlated noise detrended fluctuation analysis dfa c k peng s v buldyrev a l goldberger s havlin f sciortino m simons h e stanley nature 356 1992 168 of volatility series has been proposed to identify possible nonlinear multifractal signatures in the given empirical sample y ashkenazy p ch ivanov s havlin c k peng a l goldberger h e stanley phys rev lett 86 2001 1900 y ashkenazy s havlin p ch ivanov c k peng v schulte frohlinde h e stanley physica a 323 2003 19 t kalisky y ashkenazy s havlin phys rev e 72 2005 011913 long range volatility correlation can be an outcome of static as well as dynamical nonlinearity in order to argue in favor of dynamical nonlinearity surrogate testing is used in conjunction with volatility analysis y ashkenazy p ch ivanov s havlin c k peng a l goldberger h e stanley phys rev lett 86 2001 1900 y ashkenazy s havlin p ch ivanov c k peng v schulte frohlinde h e stanley physica a 323 2003 19 t kalisky y ashkenazy s havlin phys rev e 72 2005 011913 in this brief communication surrogate testing of volatility series from long range correlated monofractal noise and their static invertible nonlinear transforms is investigated long range correlated noise is generated from fractional auto regressive integrated moving average farima 0 d 0 with gaussian and non gaussian innovations we show significant deviation in the scaling behavior between the empirical sample and the surrogate counterpart at large time scales in the case of farima 0 d 0 with non gaussian innovations whereas no such discrepancy was observed in the case of gaussian innovations the results encourage cautious interpretation of surrogate analysis of volatility series in the presence of non gaussian innovations

a q operator for the quantum transfer matrix baxter s concept of a q operator is generalized to the quantum transfer matrix of the xxz spin chain by employing the representation theory of quantum groups the spectrum of this q operator is discussed and novel functional relations which describe the finite temperature regime of the xxz spin chain are derived for a non vanishing magnetic field the previously known bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies for vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six vertex transfer matrix at roots of unity the quantum classical crossover is also discussed in terms of the eigenvalues of the q operator for a few special examples

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

magnitude and sign scaling in power law correlated time series a time series can be decomposed into two sub series a magnitude series and a sign series here we analyze separately the scaling properties of the magnitude series and the sign series using the increment time series of cardiac interbeat intervals as an example we 6nd that time series having identical distributions and long range correlation properties can exhibit quite di8erent temporal organizations of the magnitude and sign sub series from the cases we study it follows that the long range correlations in the magnitude series indicate nonlinear behavior speci6cally our results suggest that the correlation exponent of the magnitude series is a monotonically increasing function of the multifractal spectrum width of the original series on the other hand the sign series mainly relates to linear properties of the original series we also show that the magnitude and sign series of the heart interbeat interval series can be used for diagnosis purposes c

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

random cascades on wavelet dyadic trees we introduce a new class of random fractal functions using the orthogonal wavelet transform these functions are built recursively in the space scale half plane of the orthogonal wavelet transform cascading from an arbitrary given large scale towards small scales to each random fractal function corresponds a random cascading process referred to as a w cascade on the dyadic tree of its orthogonal wavelet coefficients we discuss the convergence of these cascades and the regularity of the so obtained random functions by studying the support of their singularity spectra then we show that very different statistical quantities such as correlation functions on the wavelet coefficients or the wavelet based multifractal formalism partition functions can be used to characterize very precisely the underlying cascading process we illustrate all our results on various numerical examples

detecting long range correlations with detrended fluctuation analysis abstract we examine the detrended fluctuation analysis dfa which is a well established method for the detection of long range correlations in time series we show that deviations from scaling which appear at small time scales become stronger in higher orders of dfa and suggest a modified dfa method to remove them the improvement is necessary especially for short records that are affected by non stationarities furthermore we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of dfa

hidden grassmann structure in the xxz model for the critical xxz model we consider the space mathcal w alpha of operators which are products of local operators with a disorder operator we introduce two anti commutative families of operators bf b zeta bf c zeta which act on mathcal w alpha these operators are constructed as traces over representations of the q oscillator algebra in close analogy with baxter s q operators we show that the vacuum expectation values of operators in mathcal w alpha can be expressed in terms of an exponential of a quadratic form of bf b zeta bf c zeta

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

new formulae for solutions of quantum knizhnik zamolodchikov equations on level 4 we present a new form of solution to the quantum knizhnik zamolodchikov equation qkz on level 4 in a special case corresponding to the heisenberg xxx spin chain our form is equivalent to the integral representation obtained by jimbo and miwa in 1996 7 an advantage of our form is that it is reduced to the product of single integrals this fact is deeply related to a cohomological nature of our formulae our approach is also based on the deformation of hyper elliptic integrals and their main property deformed riemann bilinear relation jimbo and miwa also suggested a nice conjecture which relates solution of the qkz on level 4 to any correlation function of the xxx model this conjecture together with our form of solution to the qkz makes it possible to prove a conjecture that any correlation function of the xxx model can be expressed in terms of the riemann function at odd arguments and rational coefficients suggested in 8 9 this issue will be discussed in our next publication

fifth neighbour spin spin correlator for the anti ferromagnetic heisenberg chain we study the generating function of the spin spin correlation functions in the ground state of the anti ferromagnetic spin 1 2 heisenberg chain without magnetic field we have found its fundamental functional relations from those for general correlation functions which originate in the quantum knizhnik zamolodchikov equation using these relations we have calculated the explicit form of the generating functions up to n 6 accordingly we can obtain the spin spin correlator szjszj k up to k 5

density matrices for finite segments of heisenberg chains of arbitrary length we derive a multiple integral representing the ground state density matrix of a segment of length m of the xxz spin chain on l lattice sites which depends on l only parametrically this allows us to treat chains of arbitrary finite length specializing to the isotropic limit of the xxx chain we show for small m that the multiple integrals factorize we conjecture that this property holds for arbitrary m and suggest an exponential formula for the density matrix which involves only a double cauchy type integral in the exponent we demonstrate the efficiency of our formula by computing the next to nearest neighbour zz correlation function for chain lengths ranging from two to macroscopic numbers

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

improved surrogate data for nonlinearity tests current tests for nonlinearity compare a time series to the null hypothesis of a gaussian linear stochastic process for this restricted null assumption random surrogates can be constructed which are constrained by the linear properties of the data we propose a more general null hypothesis allowing for nonlinear rescalings of a gaussian linear process we show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity an iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

weighted divisor sums and bessel function series ii abstract on page 335 in his lost notebook ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of bessel functions in each case there are three possible interpretations for the double series in an earlier paper two of the present authors proved the first identity under one possible interpretation in the present paper the second identity is proved under a similar interpretation with one additional assumption moreover under a second interpretation entirely different proofs of both identities depending on weighted or twisted divisor sums are offered the two identities are intimately connected with the classical circle and divisor problems respectively