linear systems with levy fluctuations abstract we have examined the response of multivariable linear systems to levy fluctuations the characteristic function of the response is obtained explicitly as a function of time using the characteristic function the fractional moments of the process are determined the nature of the product rule obeyed by the characteristic function of a general linear dynamical system is elucidated

a note on the ziv lempel model for compressing individual sequences corresp the ziv lempel compression algorithm is a string matching and parsing approach to data compression the symbolwise equivalent for parsing models has been defined by rissanen and langdon and gives the same ideal codelength at the same cost in coding parameters by describing the context and coding parameter for each symbol an insight is provided into how the ziv lempel method achieves compression this treatment does not employ a probabilistic source for the data string the ziv lempel method effectively counts symbol instances within parsed phrases the coding parameter for each symbolwise context is determined by cumulative count ratios the code string length increase for a symbol y following substring s under the symbolwise equivalent is the log of the ratio of node counts in subtrees s and s cdot y of the ziv lempel parsing tree to demonstrate the symbolwise equivalent of the ziv lempel algorithm we extend the work of rissanen and langdon to incomplete parse trees the result requires the proper handling of the comma when one phrase is the prefix of another phrase

diffusion in heterogeneous media diffusion processes in heterogeneous media and biological systems in particular are riddled with the difficult theoretical issue of whether the true origin of anomalous behavior is renewal or memory or a special combination of the two accounting for the possible mixture of renewal and memory sources of subdiffusion is challenging from a computational point of view as well this problem is exacerbated by the limited number of techniques available for solving fractional diffusion equations with time dependent coefficients we propose an iterative scheme for solving fractional differential equations with time dependent coefficients that is based on a parametric expansion in the fractional index we demonstrate how this method can be used to predict the long time behavior of nonautonomous fractional differential equations by studying the anomalous diffusion process arising from a mixture of renewal and memory sources

dynamical origin of memory and renewal we show that the dynamic approach to fractional brownian motion fbm establishes a link between a non poisson renewal process with abrupt jumps resetting to zero the system s memory and correlated dynamic processes whose individual trajectories keep a nonvanishing memory of their past time evolution it is well known that the recrossings of the origin by an ordinary one dimensional diffusion trajectory generates a l evy and thus renewal process of index ensuremath theta 1 2 we prove with theoretical and numerical arguments that this is the special case of a more general condition insofar as the recrossings produced by the dynamic fbm generates a l evy process with 0l ensuremath theta l1 this result is extended to produce a satisfactory model for the fluorescent signal of blinking quantum dots

fractional trajectories decorrelation versus friction abstract the fundamental connection between fractional calculus and subordination processes is explored and affords a physical interpretation of a fractional trajectory that being an average over an ensemble of stochastic trajectories heretofore what has been interpreted as intrinsic friction a form of non markovian dissipation that automatically arises from adopting the fractional calculus is shown to be a manifestation of decorrelations between trajectories we apply the general theory developed herein to the lotka volterra ecological model providing new insight into the final equilibrium state the relaxation time to achieve this state is also considered

fluctuation dissipation process without a time scale we study the influence of a dissipation process on diffusion dynamics triggered by fluctuations with long range correlations we make the assumption that the perturbation process involved is of the same kind as those recently studied numerically and theoretically with a good agreement between theory and numerical treatment as a result of this assumption the equilibrium distribution departs from the ordinary canonical distribution the distribution tails are truncated the distribution border is signaled by sharp peaks and in the weak dissipation limit the central distribution body becomes identical to a truncated l evy distribution

fractional diffusions with time varying coefficients this paper is concerned with the fractionalized diffusion equations governing the law of the fractional brownian motion bh t we obtain solutions of these equations which are probability laws extending that of bh t our analysis is based on mcbride fractional operators generalizing the hyper bessel operators l and converting their fractional power l into erdelyi kober fractional integrals we study also probabilistic properties of the random variables whose distributions satisfy space time fractional equations involving caputo and riesz fractional derivatives some results emerging from the analysis of fractional equations with time varying coefficients have the form of distributions of time changed random variables

on optimal sequential prediction for general processes this paper considers several aspects of the sequential prediction problem for unbounded nonstationary processes under pth power loss spl lscr sub p u v u v sup p 1 p spl infin in the first part it is shown that bayes prediction schemes are cesaro optimal under general conditions that cesaro optimal prediction schemes are unique in a natural sense and that cesaro optimality is equivalent to a form of weak calibration connections between calibration and stronger forms of optimality are considered extensions of the existence and uniqueness results to generalized prediction and prediction from observations with additive noise are established for binary processes it is shown that thresholding an optimal prediction scheme for the squared loss yields an optimal binary prediction scheme for the hamming loss in the second part of the paper it is shown how to construct from a countable family of prediction schemes a single composite scheme whose asymptotic performance on any suitable process dominates the performance of each member of the family the construction is based on aggregating methods for individual binary sequences using the construction some results of algoet 1994 1995 on the existence of cesaro optimal schemes for families of ergodic processes are rederived in a direct way and extended to unbounded processes

fractional correlation functions in simple viscoelastic liquids abstract we develop a hydrodynamic formulation of fractional fluctuations in a viscoelastic liquid whose longitudinal modulus is a scalar and only depends on time the method is based on the introduction of fractional time derivatives in the hydrodynamic equations due to the viscoelastic memory coupled generalized langevin equations for the fluctuations of the state variables are formulated and solved analytically for a power law memory kernel with long correlation noise the associated fractional fokker planck equation ffpe is also derived and it is shown that it exhibits a fluctuation dissipation theorem fdt the explicit analytic diffusion coefficient is calculated for power law viscoelasticity and is shown to be subdiffusive the fractional correlations for longitudinal velocity and density fluctuations are analytically calculated and used to obtain the light scattering spectrum and the intermediate scattering function of the viscoelastic fluid our model calculation predicts that the fractional effects on these properties are not small effects and might be measurable

the strong law of large numbers for sequential decisions under uncertainty combines optimization and ergodic theory to characterize the optimum long run average performance that can be asymptotically attained by nonanticipating sequential decisions let x sub t be a stationary ergodic process and suppose an action b sub t must be selected in a space spl bscr with knowledge of the t past x sub 0 spl middot spl middot spl middot x sub t 1 at the beginning of every period t spl ges 0 action b sub t will incur a loss l b sub t x sub t at the end of period t when the random variable x sub t is revealed the author proves under mild integrability conditions that the optimum strategy is to select actions that minimize the conditional expected loss given the currently available information at each step the minimum long run average loss per decision can be approached arbitrarily closely by strategies that are finite order markov and under certain continuity conditions it is equal to the minimum expected loss given the infinite past if the loss l b x is bounded and continuous and if the space spl bscr is compact then the minimum can be asymptotically attained even if the distribution of the process x sub t is unknown a priori and must be learned from experience

nonparametric inference for ergodic stationary time series the setting is a stationary ergodic time series the challenge is to construct a sequence of functions each based on only finite segments of the past which together provide a strongly consistent estimator for the conditional probability of the next observation given the infinite past ornstein gave such a construction for the case that the values are from a finite set and recently algoet extended the scheme to time series with coordinates in a polish space the present study relates a different solution to the challenge the algorithm is simple and its verification is fairly transparent some extensions to regression pattern recognition and on line forecasting are mentioned

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

adaptive minimax regression estimation over sparse l q hulls given a dictionary of mn predictors in a random design regression setting with n observations we construct estimators that target the best performance among all the linear combinations of the predictors under a sparse lq norm 0 q 1 constraint on the linear coefficients besides identifying the optimal rates of convergence our universal aggregation strategies by model mixing achieve the optimal rates simultaneously over the full range of 0 q 1 for any mn and without knowledge of the lq norm of the best linear coefficients to represent the regression function to allow model misspecification our upper bound results are obtained in a framework of aggregation of estimates a striking feature is that no specific relationship among the predictors is needed to achieve the upper rates of convergence hence permitting basically arbitrary correlations between the predictors therefore whatever the true regression function assumed to be uniformly bounded our estimators automatically exploit any sparse representation of the regression function if any to the best extent possible within the lq constrained linear combinations for any 0 q 1 a sparse approximation result in the lq hulls turns out to be crucial to adaptively achieve minimax rate optimal aggregation it precisely characterizes the number of terms needed to achieve a prescribed accuracy of approximation to the best linear combination in an lq hull for 0 q 1 it offers the insight that the minimax rate of lq aggregation is basically determined by an effective model size which is a sparsity index that depends on q mn n and the lq norm bound in an easily interpretable way based on a classical model selection theory that deals with a large number of models

sequential prediction of unbounded stationary time series a simple on line procedure is considered for the prediction of a real valued sequence the algorithm is based on a combination of several simple predictors if the sequence is a realization of an unbounded stationary and ergodic random process then the average of squared errors converges almost surely to that of the optimum given by the bayes predictor an analog result is offered for the classification of binary processes

a sandwich proof of the shannon mcmillan breiman theorem let x t be a stationary ergodic process with distribution p admitting densities p x 0 ldots x n 1 relative to a reference measure m that is finite order markov with stationary transition kernel let i m p denote the relative entropy rate then n 1 log p x 0 ldots x n 1 rightarrow i m p mathrm a s p we present an elementary proof of the shannon mcmillan breiman theorem and the preceding generalization obviating the need to verify integrability conditions and also covering the case i m p infty a sandwich argument reduces the proof to direct applications of the ergodic theorem

weakly convergent nonparametric forecasting of stationary time series the conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past several schemes are known for constructing pointwise consistent estimates but they all demand prohibitive amounts of input data we consider real valued time series and construct conditional distribution estimates that make much more efficient use of the input data the estimates are consistent in a weak sense and the question whether they are pointwise consistent is still open for finite alphabet processes one may rely on a universal data compression scheme like the lempel ziv 1978 algorithm to construct conditional probability mass function estimates that are consistent in expected information divergence consistency in this strong sense cannot be attained in a universal sense for all stationary processes with values in an infinite alphabet but weak consistency can some applications of the estimates to on line forecasting regression and classification are discussed

memory universal prediction of stationary random processes we consider the problem of one step ahead prediction of a real valued stationary strongly mixing random process xi sub i spl infin sup spl infin the best mean square predictor of x sub 0 is its conditional mean given the entire infinite past x sub i sub i spl infin sup 1 given a sequence of observations x sub 1 x sub 2 x sub n we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension for example neural networks and legendre polynomials the proposed estimators select both the model memory and the model dimension in a data driven fashion by minimizing certain complexity regularized least squares criteria when the underlying predictor function has a finite memory we establish that the proposed estimators are memory universal the proposed estimators which do not know the true memory deliver the same statistical performance rates of integrated mean squared error as that delivered by estimators that know the true memory furthermore when the underlying predictor function does not have a finite memory we establish that the estimator based on legendre polynomials is consistent

a simple randomized algorithm for sequential prediction of ergodic time series we present a simple randomized procedure for the prediction of a binary sequence the algorithm uses ideas from previous developments of the theory of the prediction of individual sequences we show that if the sequence is a realization of a stationary and ergodic random process then the average number of mistakes converges almost surely to that of the optimum given by the bayes predictor the desirable finite sample properties of the predictor are illustrated by its performance for markov processes in such cases the predictor exhibits near optimal behavior even without knowing the order of the markov process prediction with side information is also considered

the weighted majority algorithm we study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials with a prediction to be made in each and the goal of the learner is to make few mistakes we are interested in the case where the learner has reason to believe that one of some pool of known algorithms will perform well but the learner does not know which one a simple and effective method based on weighted voting is introduced for constructing a compound algorithm in such a circumstance we call this method the weighted majority algorithm we show that this algorithm is robust in the presence of errors in the data we discuss various versions of the weighted majority algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool for example given a sequence of trials if there is an algorithm in the pool a that makes at most m mistakes then the weighted majority algorithm will make at most c log a m mistakes on that sequence where c is fixed constant

can intermittent long range jumps of a random walker compensate for lethargy we study the dynamics of a lazy random walker who is inactive for extended times and tries to make up for her laziness with very large jumps she remains in a condition of rest for a time derived from a waiting time distribution with w 1 the most convenient choice to make up for the random walker laziness would be to select 2 would produce levy flights with scaling 1 1 and consequently super diffusion according to the sparre andersen theorem the distribution density of the first times to go from xa to xb xa has the inverse power law form with fpt sa 1 5 we find the surprising result that there exists a region of the phase space w with w sa and the lazy walker compensates for her laziness there also exists an extended region breaking the sparre andersen theorem where the lazy runner cannot compensate for her laziness we make conjectures concerning the possible relevance of this mathematical prediction supported by numerical calculations for the problem of animal foraging

near optimal rates for limited delay universal lossy source coding we consider the problem of limited delay lossy coding of individual sequences here the goal is to design fixed rate compression schemes to minimize the normalized expected distortion redundancy relative to a reference class of coding schemes measured as the difference between the average distortion of the algorithm and that of the best coding scheme in the reference class in compressing a sequence of length t the best schemes available in the literature achieve an o t 1 3 normalized distortion redundancy relative to finite reference classes of limited delay and limited memory and the same redundancy is achievable up to logarithmic factors when the reference class is the set of scalar quantizers it has also been shown that the distortion redundancy is at least of order 1 t in the latter case and the lower bound can easily be extended to sufficiently powerful possibly finite reference coding schemes in this paper we narrow the gap between the upper and lower bounds and give a compression scheme whose normalized distortion redundancy is o ln t t relative to any finite class of reference schemes only a logarithmic factor larger than the lower bound the method is based on the recently introduced shrinking dartboard prediction algorithm a variant of exponentially weighted average prediction the algorithm is also extended to the problem of joint source channel coding over a known stochastic noisy channel and to the case when side information is also available to the decoder the wyner ziv setting the same improvements are obtained for these settings as in the case of a noiseless channel our method is also applied to the problem of zero delay scalar quantization where o ln t 1 t normalized distortion redundancy is achieved relative to the infinite class of scalar quantizers of a given rate almost achieving the known lower bound of order 1 t the computationally efficient algorithms known for scalar quantization and the wyner ziv setting carry over to our improved coding schemes presented in this paper

comment on fractional fokker planck equation with space and time dependent drift and diffusion the purpose of this paper is to correct errors presented recently in the paper lv et al j stat phys 149 619 628 2012 where the authors analyzed fractional fokker planck equation ffpe with space time dependent drift f x t f x f t and diffusion d x t d x tilde d t coefficients in the factorized form we show an important drawback in the derivation of the stochastic representation of ffpe presented in the aforementioned paper which makes the whole proof wrong moreover we present a correct proof of their result in even more general case when both drift and diffusion can have any not necessarily factorized form

ergodic properties of fractional brownian langevin motion we investigate the time average mean square displacement overline ensuremath delta 2 mathbf x t mathbf ensuremath int 0 t ensuremath ensuremath delta x t ensuremath ensuremath delta ensuremath x t ensuremath 2 d t ensuremath t ensuremath ensuremath delta for fractional brownian langevin motion where x t is the stochastic trajectory and ensuremath delta is the lag time unlike the previously investigated continuous time random walk model overline ensuremath delta 2 converges to the ensemble average x 2 ensuremath sim t 2h in the long measurement time limit the convergence to ergodic behavior is slow however and surprisingly the hurst exponent h frac 3 4 marks the critical point of the speed of convergence when hl frac 3 4 the ergodicity breaking parameter e b mathbf overline ensuremath delta 2 mathbf x t mathbf mathbf 2 ensuremath overline ensuremath delta 2 mathbf x t mathbf 2 mathbf overline ensuremath delta 2 mathbf x t mathbf 2 ensuremath sim k h ensuremath delta t ensuremath 1 when h frac 3 4 e b ensuremath sim frac 9 16 mathrm ln phantom rule 0 2em 0ex t ensuremath delta t ensuremath 1 and when frac 3 4 lhl1 e b ensuremath sim k h ensuremath delta 4 ensuremath 4h t 4h ensuremath 4 in the ballistic limit h ensuremath rightarrow 1 ergodicity is broken and e b ensuremath sim 2 the critical point h frac 3 4 is marked by the divergence of the coefficient k h fractional brownian motion as a model for recent experiments of subdiffusion of mrna in the cell is briefly discussed and a comparison with the continuous time random walk model is made

universal linear prediction by model order weighting a common problem that arises in adaptive filtering autoregressive modeling or linear prediction is the selection of an appropriate order for the underlying linear parametric model we address this problem for linear prediction but instead of fixing a specific model order we develop a sequential prediction algorithm whose sequentially accumulated average squared prediction error for any bounded individual sequence is as good as the performance attainable by the best sequential linear predictor of order less than some m this predictor is found by transforming linear prediction into a problem analogous to the sequential probability assignment problem from universal coding theory the resulting universal predictor uses essentially a performance weighted average of all predictors for model orders less than m efficient lattice filters are used to generate the predictions of all the models recursively resulting in a complexity of the universal algorithm that is no larger than that of the largest model order examples of prediction performance are provided for autoregressive and speech data as well as an example of adaptive data equalization

a context quantization approach to universal denoising we revisit the problem of denoising a discrete time continuous amplitude signal corrupted by a known memoryless channel by modifying our earlier approach to the problem we obtain a scheme that is much more tractable than the original one and at the same time retains the universal optimality properties the universality refers to the fact that the proposed denoiser asymptotically with increasing block length of the data achieves the performance of an optimum denoiser that has full knowledge of the distribution of a source generating the underlying clean sequence the only restriction being that the distribution is stationary the optimality in a sense we will make precise of the denoiser also holds in the case where the underlying clean sequence is unknown and deterministic and the only source of randomness is in the noise the schemes involve a simple preprocessing step of quantizing the noisy symbols to generate quantized contexts the quantized context value corresponding to each sequence component is then used to partition the unquantized symbols into subsequences a universal symbol by symbol denoiser for unquantized sequences is then separately employed on each of the subsequences we identify a rate at which the context length and quantization resolution should be increased so that the resulting scheme is universal the proposed family of schemes is computationally attractive with an upper bound on complexity which is independent of the context length and the quantization resolution initial experimentation seems to indicate that these schemes are not only superior from a computational viewpoint but also achieve better denoising in practice

universal modeling and coding the problems arising in the modeling and coding of strings for compression purposes are discussed the notion of an information source that simplifies and sharpens the traditional one is axiomatized and adaptive and nonadaptive models are defined with a measure of complexity assigned to the models a fundamental theorem is proved which states that models that use any kind of alphabet extension are inferior to the best models using no alphabet extensions at all a general class of so called first in first out fifo arithmetic codes is described which require no alphabet extension devices and which therefore can be used in conjunction with the best models because the coding parameters are the probabilities that define the model their design is easy and the application of the code is straightforward even with adaptively changing source models

weakly universally consistent forecasting of stationary and ergodic time series static forecasting of stationary and ergodic time series is considered i e inference of the conditional expectation of the response variable at time zero given the infinite past it is shown that the mean squared error of a combination of suitably defined localized least squares estimates converges to zero for all distributions where the response variable is square integrable

long term prediction intervals of time series we consider the problem of predicting aggregates or sums of future values of a process based on its past values in contrast with the conventional prediction problem in which one predicts a future value given past values of the process in our setting the number of aggregates can go to infinity with respect to the number of available observations consistency and bahadur representations of the prediction estimators are established a simulation study is carried out to assess the performance of different prediction estimators

on linear groups of degrees at most p 1 abstract in this paper we determine the structure of complex linear groups g of degree at most p 1 where p is a t i sylow p subgroup of g

scaled brownian motion as a mean field model for continuous time random walks we consider scaled brownian motion sbm a random process described by a diffusion equation with explicitly time dependent diffusion coefficient d t d0t 1 batchelor s equation which for 1 is often used for fitting experimental data for subdiffusion of unclear genesis we show that this process is a close relative of subdiffusive continuous time random walks and describes the motion of the rescaled mean position of a cloud of independent walkers it shares with subdiffusive continuous time random walks its nonstationary and nonergodic properties the nonergodicity of sbm does not however go hand in hand with strong difference between its different realizations its heterogeneity ergodicity breaking parameter tends to zero for long trajectories

asymptotic optimality and asymptotic equipartition properties of log optimum investment on montre comment un investisseur peut repartir ses fonds entre plusieurs opportunites d investissement pour maximiser le taux de croissance de son capital

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

sequential quantile prediction of time series motivated by a broad range of potential applications we address the quantile prediction problem of real valued time series we present a sequential quantile forecasting model based on the combination of a set of elementary nearest neighbor type predictors called experts and show its consistency under a minimum of conditions our approach builds on the methodology developed in recent years for prediction of individual sequences and exploits the quantile structure as a minimizer of the so called pinball loss function we perform an in depth analysis of real world data sets and show that this nonparametric strategy generally outperforms standard quantile prediction methods

gambling using a finite state machine sequential gambling schemes in which the amount wagered on the future outcome is determined by a finite state fs machine are defined and analyzed it is assumed that the fs machine determines the fraction of the capital wagered at each time instance i on the outcome at the next time instance i 1 and that wagers are paid at even odds the maximal capital achieved by any fs machine is found and its dependence on an empirical entropy measure h sup fs x defined as the finite state complexity of x is shown a specific gambling scheme is then proposed based on the lempel ziv method for universal compression the capital gained by this method is found and it is observed that asymptotically its exponential growth rate dominates the experimental growth rate achieved by gambling using any fs machine furthermore this specific scheme suggests a class of gambling methods based on a class of variable to variable length lossless compression methods in which the capital is doubled for every bit compressed these results emphasize the relation between gambling and data compression

levy flights from a continuous time process l evy flight dynamics can stem from simple random walks in a system whose operational time number of steps n typically grows superlinearly with physical time t thus this process is a kind of continuous time random walk ctrw dual to the typical scher montroll model in which n grows sublinearly with t models in which l evy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force which is shown to be nonlinear on the other hand the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern and leads to a normal boltzmann distribution mixed models describing normal ctrw s in superlinear operational time and l evy flights under the operational time of subdiffusive ctrw s lead to a paradoxical diffusive behavior similar to the one found in transport on polymer chains the relaxation to the boltzmann distribution in such models is slow and asymptotically follows a power law

self similar gaussian processes for modeling anomalous diffusion we study some gaussian models for anomalous diffusion which include the time rescaled brownian motion two types of fractional brownian motion and models associated with fractional brownian motion based on the generalized langevin equation gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean square displacement to vary with t ensuremath alpha 0l ensuremath alpha l2 however these processes have different properties thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism although the two versions of fractional brownian motion and time rescaled brownian motion all have the same probability distribution function the slepian theorem can be used to compare their first passage time distributions which are different finally in order to model anomalous diffusion with a variable exponent ensuremath alpha t it is necessary to consider the multifractional extensions of these gaussian processes

universal prediction of individual sequences the problem of predicting the next outcome of an individual binary sequence using finite memory is considered the finite state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite state fs predictor it is proven that this fs predictability can be achieved by universal sequential prediction schemes an efficient prediction procedure based on the incremental parsing procedure of the lempel ziv data compression algorithm is shown to achieve asymptotically the fs predictability some relations between compressibility and predictability are discussed and the predictability is proposed as an additional measure of the complexity of a sequence

a fast direct method for block triangular toeplitz like with tri diagonal block systems from time fractional partial differential equations in this paper we study the block lower triangular toeplitz like with tri diagonal blocks system which arises from the time fractional partial differential equation existing fast numerical solver e g fast approximate inversion method cannot handle such linear system as the main diagonal blocks are different the main contribution of this paper is to propose a fast direct method for solving this linear system and to illustrate that the proposed method is much faster than the classical block forward substitution method for solving this linear system our idea is based on the divide and conquer strategy and together with the fast fourier transforms for calculating toeplitz matrix vector multiplication the complexity needs o m n log 2 m arithmetic operations where m is the number of blocks the number of time steps in the system and n is the size number of spatial grid points of each block numerical examples from the finite difference discretization of time fractional partial differential equations are also given to demonstrate the efficiency of the proposed method

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function