growth of rounding errors and repetitivity of large eddy simulations this paper studies the propagation of rounding errors in large eddy simulation and shows that instantaneous flowfields produced by large eddy simulation are partially controlled by these rounding errors and depend on multiple parameters number of processors used for parallel simulation even in an explicit code changes in initial conditions even of the order of machine accuracy machine precision simple double or quadruple etc using a laminar poiseuille pipe flow a fully developed turbulent channel flow and a complex burner geometry as test cases results show that only turbulent flows exhibit a high sensitivity to these parameters these results confirm that large eddy simulation reflects the true nature of turbulence insofar as it may exponentially amplify infinitely small perturbations on initial conditions in time however they highlight an often overlooked limitation of large eddy simulation in terms of validation and prediction of unsteady phenomena

a fully discrete stable and conservative summation by parts formulation for deforming interfaces we introduce an interface coupling procedure for hyperbolic problems posed on time dependent curved multi domains first we transform the problem from cartesian to boundary conforming curvilinear coordinates and apply the energy method to derive well posed and conservative interface conditions next we discretize the problem in space and time by employing finite difference operators that satisfy a summation by parts rule the interface condition is imposed weakly using a penalty formulation we show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface while both stability and conservation conditions are respected the developed techniques are illustrated by performing numerical experiments on the linearized euler equations and the maxwell equations the results corroborate the stability and accuracy of the fully discrete approximations

scaling behaviour of entropy estimates entropy estimation of information sources is highly non trivial for symbol sequences with strong long range correlations the rabbit sequence related to the symbolic dynamics of the nonlinear circle map at the critical point as well as the logistic map at the feigenbaum point is known to produce long memory tails for both dynamical systems the scaling behaviour of the block entropy of order n has been shown to increase log n in contrast to such probabilistic concepts we investigate the scaling behaviour of certain non probabilistic entropy estimation schemes suggested by lempel and ziv lz in the context of algorithmic complexity and data compression these are applied in a sequential manner with the scaling variable being the length n of the sequence we determine the scaling law for the lz entropy estimate applied to the case of the critical circle map and the logistic map at the feigenbaum point in a binary partition

comparison of detrending methods for fluctuation analysis abstract we examine several recently suggested methods for the detection of long range correlations in data series based on similar ideas as the well established detrended fluctuation analysis dfa in particular we present a detailed comparison between the regular dfa and two recently suggested methods the centered moving average cma method and a modified detrended fluctuation analysis mdfa we find that cma performs the same as dfa in long data with weak trends and is slightly superior to dfa in short data with weak trends when comparing standard dfa to mdfa we observe that dfa performs slightly better in almost all examples we studied we also discuss how several types of trends affect different types of dfa for weak trends in the data the new methods are comparable with dfa in these respects however if the functional form of the trend in data is not a priori known dfa remains the method of choice only a comparison of dfa results using different detrending polynomials yields full recognition of the trends a comparison with independent methods is recommended for proving long range correlations

time and frequency domain characteristics of detrending operation based scaling analysis exact dfa and dma frequency responses we develop a general framework to study the time and frequency domain characteristics of detrending operation based scaling analysis methods such as detrended fluctuation analysis dfa and detrending moving average dma analysis in this framework using either the time or frequency domain approach the frequency responses of detrending operations are calculated analytically although the frequency domain approach based on conventional linear analysis techniques is only applicable to linear detrending operations the time domain approach presented here is applicable to both linear and nonlinear detrending operations furthermore using the relationship between the time and frequency domain representations of the frequency responses the frequency domain characteristics of nonlinear detrending operations can be obtained based on the calculated frequency responses it is possible to establish a direct connection between the root mean square deviation of the detrending operation based scaling analysis and the power spectrum for linear stochastic processes here by applying our methods to dfa and dma including higher order cases exact frequency responses are calculated in addition we analytically investigate the cutoff frequencies of dfa and dma detrending operations and show that these frequencies are not optimally adjusted to coincide with the corresponding time scale

establishing a direct connection between detrended fluctuation analysis and fourier analysis to understand methodological features of the detrended fluctuation analysis dfa using a higher order polynomial fitting we establish the direct connection between dfa and fourier analysis based on an exact calculation of the single frequency response of the dfa the following facts are shown analytically 1 in the analysis of stochastic processes exhibiting a power law scaling of the power spectral density psd s f f a higher order detrending in the dfa has no adverse effect in the estimation of the dfa scaling exponent which satisfies the scaling relation 1 2 2 the upper limit of the scaling exponents detectable by the dfa depends on the order of polynomial fit used in the dfa and is bounded by m 1 where m is the order of the polynomial fit 3 the relation between the time scale in the dfa and the corresponding frequency in the psd are distorted depending on both the order of the dfa and the frequency dependence of the psd we can improve the scale distortion by introducing the corrected time scale in the dfa corresponding to the inverse of the frequency scale in the psd in addition our analytical approach makes it possible to characterize variants of the dfa using different types of detrending as an application properties of the detrending moving average algorithm are discussed

fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains a time dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered by applying the energy method well posed boundary conditions for the continuous problem are derived summation by parts sbp operators for the space and time discretization together with a weak imposition of boundary and initial conditions using simultaneously approximation terms sats lead to a provable fully discrete energy stable conservative finite difference scheme we show how to construct a time dependent sat formulation that automatically imposes boundary conditions when and where they are required we also prove that a uniform flow field is preserved i e the numerical geometric conservation law ngcl holds automatically by using sbp sat in time and space the developed technique is illustrated by considering an application using the linearized euler equations the sound generated by moving boundaries numerical calculations corroborate the stability and accuracy of the new fully discrete approximations

convergence of the compact finite difference method for second order elliptic equations abstract in this paper we give a fourth order compact finite difference scheme for the general forms of two point boundary value problems and two dimensional elliptic partial differential equations pde s by decomposing the coefficient matrix into a sum of several matrixes after we discretize the original problems we can obtain a lower bound for the smallest eigenvalue of the coefficient matrix thus we prove the compact finite difference scheme converges with the fourth order of accuracy to solve the discretized block tri diagonal matrix equations of the two dimensional elliptic pde s we develop an efficient iterative method full multigrid method in our numerical experiments we compare the compact finite difference method with the finite element method and crank nicolson finite difference method the results show that the compact finite difference scheme is a highly efficient and accurate method

from semidiscrete to fully discrete stability of runge kutta schemes by the energy method the integration of semidiscrete approximations for time dependent problems is encountered in a variety of applications the runge kutta rk methods are widely used to integrate the ode systems which arise in this context resulting in large ode systems called methods of lines these methods of lines are governed by possibly ill conditioned systems with a growing dimension consequently the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading instead we present here a stability analysis of rk methods for well posed semidiscrete approximations based on a general energy method we review the stability question for such rk approximations and highlight its intricate dependence on the growing dimension of the problem in particular we prove the strong stability of general fully discrete rk methods governed by coercive approximations we conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection diffusion equations with variable coefficients a straightforward implementation of our results verify the strong stability of rk methods for local finite difference schemes as well as global spectral approximations since our approach is based on the energy method which is carried in the physical space and since it avoids the von neumann analysis which is carried in the dual fourier space we are able to easily adapt additional extensions due to nonperiodic boundary conditions general geometries etc

a class of energy stable high order finite difference interface schemes suitable for adaptive mesh refinement of hyperbolic problems we present a class of energy stable high order finite difference interface closures for grids with step resolution changes these grids are commonly used in adaptive mesh refinement of hyperbolic problems the interface closures are such that the global accuracy of the numerical method is that of the interior stencil the summation by parts property is built into the stencil construction and implies asymptotic stability by the energy method while being non dissipative we present one dimensional closures for fourth order explicit and compact pade type finite differences tests on the scalar one and two dimensional wave equations the one dimensional navier stokes solution of a shock and two dimensional inviscid compressible vortex verify the accuracy and stability of this class of methods

evaluating scaled windowed variance methods for estimating the hurst coefficient of time series three scaled windowed variance methods standard linear regression detrended and brdge detrended for estimating the hurst coefficient h are evaluated the hurst coefficient with 0 h 1 characterizes self similar decay in the time series autocorrelation function the scaled windowed variance methods estimate h for fractional brownian motion fbm signals which are cumulative sums of fractional gaussian noise fgn signals for all three methods both the bias and standard deviation of estimates are less than 0 05 for series having n 2 9 points estimates for short series n 2 8 are unreliable to have a 0 95 probability of distinguishing between two signals with true h differing by 0 1 more than 2 15 points are needed all three methods proved more reliable based on bias and variance of estimates than hurst s rescaled range analysis periodogram analysis and autocorrelation analysis and as reliable as dispersional analysis the latter methods can only be applied to fgn or differences of fbm while the scaled windowed variance methods must be applied to fbm or cumulative sums of fgn

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

optimized compact difference based finite volume schemes for linear wave phenomena high order compact difference based finite volume schemes are developed analyzed and implemented for linear wave propagation phenomena with particular emphasis on computational electromagnetics in the time domain the formulation combines the primitive function approach with five point spatially sixth and fourth order methods optimization in the semi discrete case is achieved by minimizing dispersion and isotropy error the fully discrete scheme is examined by adopting the classical fourth order runge kutta technique stability bounds are established and the coefficients for the spatial discretization are readjusted for optimum performance the scheme is then formally extended to multiple dimensions consistent boundary conditions are presented for the reconstruction operator as well as flux specification several calculations in one and three dimensions confirm the properties of the method

prediction and entropy of nonlinear dynamical systems and symbolic sequences with lro abstract following shannon we introduce higher order entropies and derive dynamic entropies the nth order dynamic entropy conditional entropy is a measure of the uncertainty of the next state which follows after the observation of n foregoing states the asymptotic behaviour of the dynamic entropies at large n is studied for several nonlinear model systems and for symbolic sequences with long range order lro for example we investigate 1d maps texts dna strings and time series it is shown that the existence of long correlations improves the possibility of predictions characteristic scaling laws for the higher order shannon entropies and the conditional entropies are derived and a new interpolation formula is tested finally instead of the dynamic entropies which yield mean values of the uncertainty predictability we investigate the local values of the uncertainty predictability

energy stable numerical methods for hyperbolic partial differential equations using overlapping domain decomposition overlapping domain decomposition methods otherwise known as overset grid or chimera methods are useful for simplifying the discretization of partial differential equations in or around complex geometries though in wide use such methods are prone to numerical instability unless numerical diffusion or some other form of regularization is used especially for higher order methods to address this shortcoming high order provably energy stable overlapping domain decomposition methods are derived for hyperbolic initial boundary value problems the overlap is treated by splitting the domain into pieces and using generalized summation by parts derivative operators and polynomial interpolation new implicit and explicit operators are derived that do not require regularization for stability in the linear limit applications to linear and nonlinear problems in one and two dimensions are presented where it is found the explicit operators are preferred to the implicit ones

highly accurate compact mixed methods for two point boundary value problems finite difference method and finite element method are widely used partly for their simplicity though these methods can obtain first order or second order of accuracy in this paper we give two highly accurate fourth order and sixth order of accuracy respectively while still quite simple schemes for two point boundary value problems we call them compact mixed methods since they can obtain numerical solutions for unknown function and its first derivative simultaneously to yield best numerical solutions we combine our compact mixed schemes with a direct solver and an iterative solver then we employ fourier method to analyze differencing errors shows that our compact mixed formulae are closer to the true wavenumber numerical experiments show compact mixed schemes are very efficient and highly accurate techniques

strict stability of high order compact implicit finite difference schemes temporal or strict stability of approximation to pdes is much more difficult to achieve than the classical lax stability in this paper we present a class of finite difference schemes for hyperbolic initial boundary value problems in one and two space dimensions that possess the property of strict stability the approximations are constructed so that all eigenvalues of corresponding differentiation matrix have a nonpositive real part boundary conditions are imposed by using penalty like terms fourth and sixth order compact implicit finite difference schemes are constructed and analyzed computational efficacy of the approach is corroborated by a series of numerical tests in 1 d and 2 d scalar problems

a cartesian based embedded geometry technique with adaptive high order finite differences for compressible flow around complex geometries an immersed boundary methodology to solve the compressible navier stokes equations around complex geometries in cartesian fluid dynamics solvers is described the objective of the new approach is to enable smooth reconstruction of pressure and viscous stresses around the embedded objects without spurious numerical artifacts a standard level set represents the boundary of the object and defines a fictitious domain into which the flow fields are smoothly extended boundary conditions on the surface are enforced by an approach inspired by analytic continuation each fluid field is extended independently constrained only by the boundary condition associated with that field unlike most existing methods no jump conditions or explicit derivation of them from the boundary conditions are required in this approach numerical stiffness that arises when the fluid solid interface is close to grid points of the mesh is addressed by preconditioning in addition the embedded geometry technique is coupled with a stable high order adaptive discretization that is enabled around the object boundary to enhance resolution the stencils used to transition the order of accuracy of the discretization are derived using the summation by parts technique that ensures stability applications to shock reflections shock ramp interactions and supersonic and low mach number flows over two and three dimensional geometries are presented

a dual consistent summation by parts formulation for the linearized incompressible navier stokes equations posed on deforming domains in this article well posedness and dual consistency of the linearized incompressible navier stokes equations posed on time dependent spatial domains are studied to simplify the derivation of the

on the outflow conditions for spectral solution of the viscous blunt body problem the purpose of this paper is to study and identify suitable outflow boundary conditions for the numerical simulation of viscous supersonic hypersonic flow over blunt bodies governed by the compressible navier stokes equations with an emphasis motivated primarily by the use of spectral methods without any filtering the subsonic supersonic composition of the outflow boundary requires a dual boundary treatment for well posedness all compatibility relations modified to undertake the hyperbolic parabolic behaviour of the governing equations are used for the supersonic part of the outflow regarding the unknown downstream information in the subsonic region different subsonic outflow conditions in the sense of the viscous blunt body problem are examined a verification procedure is conducted to make out the distinctive effect of each outflow condition on the solution detailed comparisons are performed to examine the accuracy and performance of the outflow conditions considered for two model geometries of different surface curvature variations numerical simulations indicate a noticeable influence of pressure from subsonic portion to supersonic portion of the boundary layer it is demonstrated that two approaches for imposing subsonic outflow conditions namely 1 extrapolating all flow variables and 2 extrapolation of pressure along with using proper compatibility relations are more suitable than the others for accurate numerical simulation of viscous high speed flows over blunt bodies using spectral collocation methods

possible biomechanical origins of the long range correlations in stride intervals of walking when humans walk the time duration of each stride varies from one stride to the next these temporal fluctuations exhibit long range correlations it has been suggested that these correlations stem from higher nervous system centers in the brain that control gait cycle timing existing proposed models of this phenomenon have focused on neurophysiological mechanisms that might give rise to these long range correlations and generally ignored potential alternative mechanical explanations we hypothesized that a simple mechanical system could also generate similar long range correlations in stride times we modified a very simple passive dynamic model of bipedal walking to incorporate forward propulsion through an impulsive force applied to the trailing leg at each push off push off forces were varied from step to step by incorporating both sensory and motor noise terms that were regulated by a simple proportional feedback controller we generated 400 simulations of walking with different combinations of sensory noise motor noise and feedback gain the stride time data from each simulation were analyzed using detrended fluctuation analysis to compute a scaling exponent a this exponent quantified how each stride interval was correlated with previous and subsequent stride intervals over different time scales for different variations of the noise terms and feedback gain we obtained short range correlations 1 0 our results indicate that a simple biomechanical model of walking can generate long range correlations and thus perhaps these correlations are not a complex result of higher level neuronal control as has been previously suggested

matched interface and boundary mib for the implementation of boundary conditions in high order central finite differences high order central finite difference schemes encounter great difficulties in implementing complex boundary conditions this paper introduces the matched interface and boundary mib method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high order central finite difference schemes to attain arbitrarily high order the mib method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions the proposed approach is extensively validated via boundary value problems initial boundary value problems eigenvalue problems and high order differential equations successful implementations are given to not only dirichlet neumann and robin boundary conditions but also more general ones such as multiple boundary conditions in high order differential equations and time dependent boundary conditions in evolution equations detailed stability analysis of the mib method is carried out the mib method is shown to be able to deliver high order accuracy while maintaining the same or similar stability conditions of the standard high order central difference approximations the application of the proposed mib method to the boundary treatment of other non standard high order methods is also considered

nondissipative and energy stable high order finite difference interface schemes for 2 d patch refined grids a class of finite difference interface schemes suitable for two dimensional cell centered grids with patch refinement and step changes in resolution is presented grids of this type are generated by adaptive mesh refinement methods according to resolution needs dictated by the physics of the problem being modeled for these grids coarse and fine nodes are not aligned at the mesh interfaces resulting in hanging nodes three distinct geometries are identified at the interfaces of a domain with interior patch refinement edges concave corners and convex corners asymptotic stability in time of the numerical scheme is achieved by imposing a summation by parts condition on the interface closure which is thus also nondissipative interface stencils corresponding to an explicit fourth order finite difference scheme are presented for each geometry to preserve stability a reduction in local accuracy is required at the corner geometries it is also found that no second order accurate solution exists that satisfies the summation by parts condition tests using the 2 d scalar advection equation and an inviscid compressible vortex support the stability and accuracy of these stencils for both linear and nonlinear problems

entropy of the symbolic sequence for critical circle maps the rabbit sequence which is related to the golden mean and the fibonacci numbers is a self similar infinite sequence of 0 s and 1 s that occurs in a variety of contexts in physics and mathematics for instance it represents the symbolic dynamics of the nonlinear circle map at the transition from periodicity to chaos and it appears in mathematical models of quasicrystals here the block entropy for the rabbit sequence is derived analytically it has already been argued that the sequence exhibits long range order our results confirm this conjecture the entropy per bit a direct measure for the long range order in symbolic sequences decays logarithmically the same behavior has been found in a heuristic way for the symbolic dynamics generated by the logistic map at the feigenbaum point

recursive procedures for measuring disorder in non periodic sequences abstract recursive procedures are found for calculating the kth order entropy of non periodic sequences these entropies are used as measures of disorder of the sequences and we treat the fibonacci sequence and generalizations the thue morse sequence together with generalizations and the period doubling sequence a discussion of the relative ordering of the thue morse and the fibonacci sequences is given and it is found too simplistic to use any one measure for comparison

high order non uniform grid schemes for numerical simulation of hypersonic boundary layer stability and transition the direct numerical simulation of receptivity instability and transition of hypersonic boundary layers requires high order accurate schemes because lower order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields the main limiting factor in the application of high order schemes to practical boundary layer flow problems is the numerical instability of high order boundary closure schemes on the wall this paper presents a family of high order non uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary layer transition by using an appropriate grid stretching and clustering grid points near the boundary high order schemes with stable boundary closures can be obtained the order of the schemes ranges from first order at the lowest to the global spectral collocation method at the highest the accuracy and stability of the new high order numerical schemes is tested by numerical simulations of the linear wave equation and two dimensional incompressible flat plate boundary layer flows the high order non uniform grid schemes up to the 11th order are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge the steady and unsteady results show that the new high order schemes are stable and are able to produce high accuracy for computations of the nonlinear two dimensional navier stokes equations for the wall bounded supersonic flow

on conservation and stability properties for summation by parts schemes abstract we discuss conservative and stable numerical approximations in summation by parts form for linear hyperbolic problems with variable coefficients an extended setting where the boundary or interface may or may not be included in the grid is considered we prove that conservative and stable formulations for variable coefficient problems require a boundary and interface conforming grid and exact numerical mimicking of integration by parts finally we comment on how the conclusions from the linear analysis carry over to the nonlinear setting

evenly spacing in detrended fluctuation analysis abstract detrended fluctuation analysis is a widely used method which aims at assessing the level of self similarity in time series this method analyzes the diffusion properties of the signal by computing the linear regression slope in the diffusion plot representing in log log coordinates the relationship between the variability of the signal and the length of the intervals over which this variability is computed we compare in this paper the results obtained with logarithmically spaced and evenly spaced diffusion plots the study shows the substantial benefits of evenly spacing especially in the reduction of the variability of estimation

the convergence rate for difference approximations to general mixed initial boundary value problems the convergence rate is investigated for the solutions to difference schemes approximating the mixed initial boundary value problem for general systems of differential equations it is shown that if an energy estimate holds then the extra boundary conditions can be of one order lower accuracy without destroying the convergence rate expected from the approximation at inner points if the maximal order of the derivatives occurring in the boundary conditions is low enough then even lower accuracy can be permitted for the extra boundary conditions

deriving dispersional and scaled windowed variance analyses using the correlation function of discrete fractional gaussian noise methods for estimating the fractal dimension d or the related hurst coefficient h for a one dimensional fractal series include hurst s method of rescaled range analysis spectral analysis dispersional analysis and scaled windowed variance analysis which is related to detrended fluctuation analysis dispersional analysis estimates h by using the variance of the grouped means of discrete fractional gaussian noise series dfgn scaled windowed variance analysis estimates h using the mean of grouped variances of discrete fractional brownian motion dfbm series both dispersional analysis and scaled windowed variance analysis have small bias and variance in their estimates of the hurst coefficient this study demonstrates that both methods derive their accuracy from their strict mathematical relationship to the expected value of the correlation function of dfgn the expected values of the variance of the grouped means for dispersional analysis on dfgn and the mean of the grouped variance for scaled windowed variance analysis on dfbm are calculated an improved formulation for scaled windowed variance analysis is given the expected values using these analyses on the wrong kind of series dispersional analysis on dfbm and scaled windowed variance analysis on dfgn are also calculated

nonparametric entropy estimation for stationary processes and random fields with applications to english text we discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a doeblin type mixing condition the estimators are cesaro averages of longest match lengths and their consistency follows from a generalized ergodic theorem due to maker 1940 we provide examples of their performance on english text and we generalize our results to countable alphabet processes and to random fields

fractal fluctuations in spatiotemporal variables when walking on a self paced treadmill abstract this study investigated the fractal dynamic properties of stride time st stride length sl and stride speed ss during walking on a self paced treadmill stm in which the belt speed is automatically controlled by the walking speed twelve healthy young subjects participated in the study the subjects walked at their preferred walking speed under four conditions stm stm with a metronome stm met fixed speed conventional treadmill ftm and ftm with a metronome ftm met to compare the fractal dynamics between conditions the mean variability and fractal dynamics of st sl and ss were compared moreover the relationship among the variables was examined under each walking condition using three types of surrogates the mean values of all variables did not differ between the two treadmills and the variability of all variables was generally larger for stm than for ftm the use of a metronome resulted in a decrease in variability in st and ss for all conditions the fractal dynamic characteristics of ss were maintained with stm in contrast to ftm and only the fractal dynamic characteristics of st disappeared when using a metronome in addition the fractal dynamic patterns of the cross correlated surrogate results were identical to those of all variables for the two treadmills in terms of the fractal dynamic properties stm walking was generally closer to overground walking than ftm walking although further research is needed the present results will be useful in research on gait fractal dynamics and rehabilitation

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

evenly spaced detrended fluctuation analysis selecting the number of points for the diffusion plot abstract detrended fluctuation analysis dfa has become a widely used tool to examine the correlation structure of a time series and provided insights into neuromuscular health and disease states as the popularity of utilizing dfa in the human behavioral sciences has grown understanding its limitations and how to properly determine parameters is becoming increasingly important dfa examines the correlation structure of variability in a time series by computing the slope of the log s d log n diffusion plot when using the traditional dfa algorithm the timescales n are often selected as a set of integers between a minimum and maximum length based on the number of data points in the time series this produces non uniformly distributed values of n in logarithmic scale which influences the estimation of due to a disproportionate weighting of the long timescale regions of the diffusion plot recently the evenly spaced dfa and evenly spaced average dfa algorithms were introduced both algorithms compute by selecting k points for the diffusion plot based on the minimum and maximum timescales of interest and improve the consistency of estimates for simulated fractional gaussian noise and fractional brownian motion time series two issues that remain unaddressed are 1 how to select k and 2 whether the evenly spaced dfa algorithms show similar benefits when assessing human behavioral data we manipulated k and examined its effects on the accuracy consistency and confidence limits of in simulated and experimental time series we demonstrate that the accuracy and consistency of are relatively unaffected by the selection of k however the confidence limits of narrow as k increases dramatically reducing measurement uncertainty for single trials we provide guidelines for selecting k and discuss potential uses of the evenly spaced dfa algorithms when assessing human behavioral data

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

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

reliability of the walking speed and gait dynamics variables while walking on a feedback controlled treadmill the purpose of this study is to identify the reliability of walking speed and gait dynamics measured with a feedback controlled treadmill and to assess the applicability of the treadmill to gait dynamics studies the intraclass correlation coefficient icc and the standard error of measurement sem for the walking speed and the mean variability coefficient of variance cv and fractal dynamics the scaling exponent of detrended fluctuation analysis dfa of the stride time and stride length were used to evaluate the within day and between day reliability fifteen subjects walked on a feedback controlled treadmill for three trials that were each more than 10 min in length within day this protocol was repeated on another day to identify the between day reliability the results showed that all variables were consistent for within day and between day reliability icc 0 633 0 982 p 0 05 sem 0 02 0 43 the within and between day reliability of the walking speed and the mean variability and fractal dynamics for the stride time and stride length were identified good iccs and low sems for within day and between day reliability were obtained for all variables therefore it is concluded that it is possible to use a feedback controlled treadmill to the study of gait dynamics

derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation in this paper simple polynomial interpolation is used to derive arbitrarily high order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative consisting of three second derivative nodes in the interior and two on the boundary on non uniform grids boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme the high order non uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non uniform grid with appropriate stretching so that more grid points are clustered near the boundary the stability and resolution properties of the high order non uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented

estimating the information content of symbol sequences and efficient codes several variants of an algorithm for estimating shannon entropies of symbol sequences are presented they are all related to the lempel ziv algorithm 1976 1977 and to recent algorithms for estimating hausdorff dimensions the average storage and running times increase as n and nlog n respectively with the sequence length n these algorithms proceed basically by constructing efficient codes they seem to be the optimal algorithms for sequences with strong long range correlations e g natural languages an application to written english illustrates their use

analyzing exact fractal time series evaluating dispersional analysis and rescaled range methods precise reference signals are required to evaluate methods for characterizing a fractal time series here we use fgp fractional gaussian process to generate exact fractional gaussian noise fgn reference signals for one dimensional time series the average autocorrelation of multiple realizations of fgn converges to the theoretically expected autocorrelation two methods commonly used to generate fractal time series an approximate spectral synthesis ssm method and the successive random addition sra method do not give the correct correlation structures and should be abandoned time series from fgp were used to test how well several versions of rescaled range analysis r s and dispersional analysis disp estimate the hurst coefficient 0 0 9 r s detrended overestimates h for time series with h 0 7 estimates of h from all versions of disp usually have lower bias and variance than those from r s all versions of dispersional analysis disp now tested on fgp are better than we previously thought and are recommended for evaluating time series as long memory processes

a novel strategy for deriving high order stable boundary closures based on global conservation i basic formulas abstract it is well known that for initial boundary value problems one may need to decrease the order of boundary closures to ensure strict stability also called time stability of finite difference schemes for high order schemes this often reduces the global convergence rate significantly to cure this issue we derive high order accurate boundary closures that satisfy a global conservation property we show that the boundary closures cannot be both high order accurate and conservative in the l 2 inner product if the grid is equidistant therefore we derive particular sets of nonuniform solution points and quadrature weights near the domain boundaries the new schemes are high order accurate on the modified grids and conservative in the derived quadrature rules although we have no theoretical stability proof numerical eigenvalue analysis suggests that the new schemes are strictly stable up to eleventh order global accuracy for scalar linear advection equations numerical experiments with one and two dimensional linear equations corroborate the design orders of the schemes

walking speed and spatiotemporal step mean measures are reliable during feedback controlled treadmill walking however spatiotemporal step variability is not reliable the purpose of the study was to compare the effects of a feedback controlled treadmill feedbacktm to a traditional fixed speed treadmill fixedtm on spatiotemporal gait means variability and dynamics the study also examined inter session reliability when using the feedbacktm ten young adults walked on the feedbacktm for a 5 minute familiarization followed by a 16 minute experimental trial they returned within one week and completed a 5 minute familiarization followed by a 16 minute experimental trial each for feedbacktm and fixedtm conditions mean walking speed and step time length width and speed means and coefficient of variation were calculated from all experimental conditions step time length width and speed gait dynamics were analyzed using detrended fluctuation analysis mean differences between experimental trials were determined using anovas and reliability between feedbacktm sessions was determined by intraclass correlation coefficient no difference was found in mean walking speed nor spatiotemporal variables with the exception of step width between the experimental trials all mean spatiotemporal variables demonstrated good to excellent reliability between sessions while coefficient of variation was not reliable gait dynamics of step time length width and speed were significantly more persistent during the feedbacktm condition compared to fixedtm especially step speed however gait dynamics demonstrated fair to poor reliability between feedbacktm sessions when walking on the feedbacktm users maintain a consistent set point yet the gait dynamics around the mean are different when compared to walking on a fixedtm in addition spatiotemporal gait dynamics and variability may not be consistent across separate days when using the feedbacktm

temporal correlations in human locomotion recommendations for sampling rate and foot strike detection abstract human stride intervals exhibit statistically persistent fluctuations over a wide range of timescales numerous studies have examined stride interval correlations during overground and treadmill walking however the practices and procedures for pre preprocessing gait data vary considerably this paper examined how differences in sampling rate and gait event detection affected stride interval correlations measured using detrended fluctuation analysis sampling rates ranged from 30 360 hz three common kinematic event detection algorithms were compared to a force based reference lower sampling frequencies failed to capture subtle variations in gait cycle timing which reduced the strength of stride interval correlations the kinematic event detection algorithms produced slightly higher but comparable estimates of stride interval correlations relative to the force based reference guidelines and considerations for assessing stride interval correlations and the implications of these findings are discussed

strict stability of high order compact implicit finite difference schemes the role of boundary conditions for hyperbolic pdes ii abstract this paper deals with the problem of systems of hyperbolic pdes in one and two space dimensions using the theory of part i 7