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

empirical mode decomposition and correlation properties of long daily ozone records sreceived 16 november 2004 revised manuscript received 17 march 2005 published 27 may 2005d correlations for daily data of total ozone column are investigated by detrended fluctuation analysis sdfad the removal of annual periodicity does not result in a background free signal for the tropical station mauna loa in order to identify the remaining quasiperiodic constituent the relatively new method of empirical mode decomposition semdd is tested we found that the so called intrinsic mode functions do not represent real signal components of the ozone time series their amplitude modulation is very sensitive to local changes such as random data removal or smoothing tests on synthetic data further corroborate the limitations of decomposing quasiperiodic signals from noise with emd nevertheless the emd algorithm helps to identify dominating frequencies in the time series which allows to separate fluctuations from the remaining background we demonstrate that dfa analysis for the cleaned mauna loa record yields scaling comparable to a mid latitude station

multifractal detrended cross correlation analysis for two nonstationary signals we propose a method called multifractal detrended cross correlation analysis to investigate the multifractal behaviors in the power law cross correlations between two time series or higher dimensional quantities recorded simultaneously which can be applied to diverse complex systems such as turbulence finance ecology physiology geophysics and so on the method is validated with cross correlated one and two dimensional binomial measures and multifractal random walks as an example we illustrate the method by analyzing two financial time series

2 fast parallel prime number sieves a prime number sieve is an algorithm that lists all prime numbers up to a given bound n two parallel prime number sieves for an algebraic erew pram model of computation are presented and analyzed the first sieve runs in o log n time using o n log n log log n processors and the second sieve runs in o root n time using o root n processors the first sieve is optimal in the sense that it performs work o n log log n which is within a constant factor of the number of arithmetic operations used by the fastest known sequential prime number sieves however when both sieves are analyzed on the block pram model as defined by aggarwal chandra and snir it is found that the second sieve is more work efficient when communication latency is significant

fractionally integrated process with power law correlations in variables and magnitudes we propose a new stochastic process de ned by only two parameters 1 and 2 which generates power law correlations or anticorrelations in the variable increments and power law correlations in their magnitudes to account for empirical observations in a range of physical systems to demonstrate the utility of our process we consider the case with power law anticorrelations in the variable increments and power law correlations in their magnitudes to model scaling properties in heartbeat data the parameters 1 and 2 control separately the correlations in the increments and their magnitudes and we nd linear relations between each parameter and the corresponding correlation exponent

time dependent hurst exponent in financial time series abstract we calculate the hurst exponent h t of several time series by dynamical implementation of a recently proposed scaling technique the detrending moving average dma in order to assess the accuracy of the technique we calculate the exponent h t for artificial series simulating monofractal brownian paths with assigned hurst exponents h we next calculate the exponent h t for the return of high frequency tick by tick sampled every minute series of the german market we find a much more pronounced time variability in the local scaling exponent of financial series compared to the artificial ones the dma algorithm allows the calculation of the exponent h t without any a priori assumption on the stochastic process and on the probability distribution function of the random variables as happens for example in the case of the kitagawa grid and the extended kalmann filtering methods the present technique examines the local scaling exponent h t around a given instant of time this is a significant advance with respect to the standard wavelet transform or to the higher order power spectrum technique which instead operate on the global properties of the series by legendre or fourier transform of qth order moments

modular exponentiation via the explicit chinese remainder theorem fix pairwise coprime positive integers p 1 p 2 p a we propose representing integers u modulo m where m is any positive integer up to roughly p 1 p 2 p s as vectors u mod p 1 u mod p 2 u mod p s we use this representation to obtain a new result on the parallel complexity of modular exponentiation there is an algorithm for the common crcw pram that given positive integers x e and m in binary of total bit length n computes x e mod m in time o n lglgn using n o 1 processors for comparison a parallelization of the standard binary algorithm takes superlinear time adleman and kompella gave an o lg n 3 expected time algorithm using exp o n lg n processors von zur gathen gave an nc algorithm for the highly special case that m is polynomially smooth

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

power law correlated processes with asymmetric distributions motivated by the fact that many physical systems display sid power law correlations together with siid an asymmetry in the probability distribution we propose a stochastic process that can model both properties the process depends on only two parameters where one controls the scaling exponent of the power law correlations and the other controls the degree of asymmetry in the distributions leaving the correlations unaffected we apply the process to air humidity data and find that the statistical properties of the process are in a good agreement with those observed in the data

cross correlations between volume change and price change in finance one usually deals not with prices but with growth rates r defined as the difference in logarithm between two consecutive prices here we consider not the trading volume but rather the volume growth rate r the difference in logarithm between two consecutive values of trading volume to this end we use several methods to analyze the properties of volume changes r and their relationship to price changes r we analyze 14 981 daily recordings of the standard and poor s s p 500 index over the 59 year period 1950 2009 and find power law cross correlations between r and r by using detrended cross correlation analysis dcca we introduce a joint stochastic process that models these cross correlations motivated by the relationship between r and r we estimate the tail exponent of the probability density function p r r 1 for both the s p 500 index as well as the collection of 1819 constituents of the new york stock exchange composite index on 17 july 2009 as a new method to estimate we calculate the time intervals q between events where r q we demonstrate that q the average of q obeys q q we find 3 furthermore by aggregating all q values of 28 global financial indices we also observe an approximate inverse cubic law

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

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

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

computing powers in parallel fast parallel computations are presented for large powers modulo an element that has only small prime factors they work for integers and polynomials over small finite fields

nonunitary transformations and hydrodynamic limit in the one dimensional hard point gas abstract the ergodic theoretical formalism of nonunitary transformations is compared with the derivation of kinetic equations by the hydrodynamic scaling limit the one dimensional hard point system is used as an illustration both methods rest on the k mixing property of the physical systems considered we show that the specific form of the operator is important only on the microscopic time scale on the macroscopic scale it acts like a projector with broken time reversal symmetry we also discuss the connection between the kolmogorov sinai k s entropy of the model dynamical system the relative entropy production rate in the nonunitary formalism and the macroscopic entropy production rate

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

a hybrid detrending method for fractional gaussian noise abstract determining trend and implementing detrending operations are important steps in data analysis yet there is neither precise definition of trend nor any logical algorithm for extracting it in this paper we propose a hybrid detrending method hdm which is based on the empirical mode decomposition emd and the detrended fluctuation analysis dfa our method can remove the polynomial and sinusoidal trends from the fractional gaussian noise fgn background we illustrate the method by selected examples from artificial time series and climate data in comparison with existing frequency domain based detrending methods our method is a posteriori the trend defined by our method is only derived from the data further our method also preserves the scaling behavior of the original signals

an fft extension to the p 1 factoring algorithm j m pollard in 1974 presented the p integer factoring al gorithm his paper couched the algorithm in theoretical terms based upon use of fast fourier transform techniques but he was unable to say whether the method could be made practical we discuss the mathematical basis of the al gorithm and show how it can work in practice the practical implementation depends for its success upon the use of residue number systems we also present an open problem as to how the method could be made to work for the elliptic curve factoring algorithm

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method