fractional brownian motion as a nonstationary process an alternative paradigm for dna sequences article discussing the fractional brownian motion fbm as a nonstationary process and an alternative paradigm for dna sequences

stability and error estimates of local discontinuous galerkin methods with implicit explicit time marching for advection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with carefully chosen implicit explicit imex runge kutta time discretization up to third order accuracy for solving one dimensional linear advection diffusion equations in the time discretization the advection term is treated explicitly and the diffusion term implicitly there are three highlights of this work the first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the ldg methods the second is that by aid of the aforementioned relationship and the energy method we show that the imex ldg schemes are unconditionally stable for the linear problems in the sense that the time step tau is only required to be upper bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent

scaling in non stationary time series ii teen birth phenomenon abstract this paper is devoted to the problem of statistical mechanics raised by the analysis of an issue of sociological interest the teen birth phenomenon it is expected that these data are characterized by correlated fluctuations reflecting the cooperative properties of the process however the assessment of the anomalous scaling generated by these correlations is made difficult and ambiguous as well by the non stationary nature of the data that shows a clear dependence on seasonal periodicity periodic component and an average changing slowly in time slow component as well we use the detrending techniques described in the companion paper the earlier companion paper to safely remove all the biases and to derive the genuine scaling of the teen birth phenomenon

molecular evolution modeled as a fractal statistical process abstract modeling the rate of nucleotide substitutions in dna as a dichotomous stochastic process with an inverse power law correlation function describes evolution by a fractal stochastic process fsp this fsp model agrees with recent findings on the relationship between the variance and mean number of synonymous and nonsynonymous substitutions in 49 different genes in mammals that being a power law increase in the ratio of the variance to the mean the index of dispersion with the number of substitutions in a protein the probability of a given number of substitutions occuring in a time t is determined by a fractional diffusion equation whose solution is a truncated levy distribution implying that evolution is a levy process in time and yields the same functional behavior for the variance in the number of substitutions as does the fsp model in addition to obtaining these relationships the fsp model implies lognormal statistics for the index of dispersion as a function of the mean number of substitutions in a protein which is confirmed in the regression of the fsp model to data lognormal statistics suggest that molecular evolution can be viewed as a multiplicative stochastic process rather than the linear additive processes of darwinian selection and drift

optimal a priori error estimates for the hp version of the local discontinuous galerkin method for convection diffusion problems we study the convergence properties of the hp version of the local discontinuous galerkin finite element method for convection diffusion problems we consider a model problem in a one dimensional space domain we allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh width h in the polynomial degree p and in the regularity of the exact solution we identify a special numerical flux for which the estimates are optimal in both h and p the theoretical results are confirmed in a series of numerical examples

local discontinuous galerkin methods with implicit explicit multistep time marching for solving the nonlinear cahn hilliard equation abstract in this paper we develop the fully discrete local discontinuous galerkin ldg methods coupled with the implicit explicit imex multistep time marching for solving the nonlinear cahn hilliard equation to be more specific we rewrite the cahn hilliard equation in a novel form by adding and subtracting a linear term then we discretize the spatial derivatives with the ldg methods and the temporal derivative with the imex multistep method finally a series of numerical experiments are given to verify the accuracy efficiency and validness of proposed method

multifractal weighted permutation analysis based on renyi entropy for financial time series abstract this paper proposes a new method called multifractal weighted permutation analysis based on renyi entropy mfwpa to calculate generalized dimension of financial time series the generalized dimension obtained by weighted permutation process retains more amplitude information of time series and is closely related to the multifractal properties of the system the advantages of this method are verified by numerical simulations we find mfwpa has a less sensitivity to noise and captures the complexity for different parts of sequences by changing the length of sequences moreover we apply this method to investigate multifractal behaviors of different stock indices and compare it with the classical algorithm called standard multifractal analysis based on partition function sma results show that mfwpa could describe the multifractal behaviors of stock indices in detail and reflect the complexity of time series in addition generalized dimensions of shuffled series are larger than the corresponding original series as a consequence of the removed autocorrelation

fractals from genomes exact solutions of a biology inspired problem abstract this is a review of a few recent papers with some new results added after a brief biological introduction a visualization scheme of the string composition of long dna sequences in particular of bacterial complete genomes will be described this scheme leads to a class of self similar and self overlapping fractals in the limit of infinitely long constituent strings the calculation of their exact dimensions and the counting of true and redundant avoided strings at different string lengths turn out to be one and the same problem we give exact solution of the problem using two independent methods the goulden jackson cluster method in combinatorics and the method of formal language theory

diffusion on multifractals abstract we introduce a class of random fields with variable mean square regularity order defined on multifractal domains the local singularity order of these random fields then depends on the initial variable mean square regularity order and on the variable local singularity exponent of the multifractal measure defining the local dimension of the domain considered the theory is developed in a generalized framework through the covariance factorization using the tools of reproducing kernel hilbert spaces and fractional sobolev spaces of variable order

noisy levy walk analog of two dimensional dna walks for chromosomes of s cerevisiae there has been interest in the physics community in studying dna sequences from a physical or mathematical point of view the findings of some recent works suggest that the sequence of base pairs or nucleotides in dna displays power law correlations 1 2 and several controversial points have been discussed in recent years most of these works are based on a one dimensional mapping of the sequence here we study a two dimensional mapping that will be described immediately below in particular in this work we analyze the two dimensional mapping of dna sequences of the organism saccharomyces cerevisiae s c baker s yeast the first eukaryote whose complete genome has been sequenced 3 this provides us with several very large sequences corresponding to the same organism such sequences consist of a succession of four symbols a adenine g guanine t thymine and c cytosine typically the frequency of a and of t is around 0 30 and that of g and c is about 0 20 the frequency of both a1g purines and c1t pyrimidines is very close to 0 5 we will envisage these sequences as realizations of a stochastic process for its analysis we introduce a mapping into a two dimensional walk with each c g symbol we associate one step in the positive negative direction along the vertical y axis with each t a symbol we associate one step to the right left along the horizontal x axis in this way we obtain a roughly unbiased walk a typical resulting pattern is displayed by chromosome ii see fig 1 among the many quantities useful to characterize a walk the mean square displacement msd is one of the most important as it is closely related to the correlations 4 for a standard random walk rw the msd r 2 s is a linear function of the number of steps s r 2 s 52dds the proportionality constant 2 dd where d is the dimension of the space defines the diffusion coefficientd this purely linear behavior is an immediate consequence of the fact that a rw is a sum of identically distributed independent random variables the total displacement after s steps r s is the sum of

stability analysis and a priori error estimates of the third order explicit runge kutta discontinuous galerkin method for scalar conservation laws in this paper we present an analysis of the runge kutta discontinuous galerkin method for solving scalar conservation laws where the time discretization is the third order explicit total variation diminishing runge kutta method we use an energy technique to prove the mathrm l 2 norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws quasi optimal order is obtained for general numerical fluxes and optimal order is given for upwind fluxes the theoretical results are obtained for piecewise polynomials with any degree k geq1 under the standard temporal spatial cfl condition tau leq gamma h where h and tau are the element length and time step respectively and the positive constant gamma is independent of h and tau

multifractal characterization of complete genomes this paper develops a theory for characterization of dna sequences based on their measure representation the measures are shown to be random cascades generated by an infinitely divisible distribution this probability distribution is uniquely determined by the exponent function in the multifractal theory of random cascades curve fitting to a large number of complete genomes of bacteria indicates that the gamma density function provides an excellent fit to the exponent function and hence to the probability distribution of the complete genomes

dynamical model for dna sequences we address the problem of dna sequences developing a dynamical method based on the assumption that the statistical properties of dna paths are determined by the joint action of two processes one deterministic with long range correlations and the other random and ensuremath delta function correlated the generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos that are responsible for the birth of anomalous diffusion it is assumed that the deterministic process corresponds to unknown biological rules that determine the dna path whereas the noise mimics the influence of an infinite dimensional environment on the biological process under study we prove that the resulting diffusion process if the effect of the random process is neglected is an ensuremath alpha stable l evy process with 12 we also show that if the diffusion process is determined by the joint action of the deterministic and the random process the correlation effects of the deterministic dynamics are cancelled on the short range scale but show up in the long range one we denote our prescription to generate statistical sequences as the copying mistake map cmm we carry out our analysis of several dna sequences and their cmm realizations with a variety of techniques and we especially focus on a method of regression to equilibrium which we call the onsager analysis with these techniques we establish the statistical equivalence of the real dna sequences with their cmm realizations we show that long range correlations are present in exons as well as in introns but are difficult to detect since the exon dynamics is shown to be determined by the entanglement of three distinct and independent cmm s

multifractal properties of power law time sequences application to rice piles we study the properties of time sequences extracted from a self organized critical system within the framework of the mathematical multifractal analysis to this end we propose a fixed mass algorithm well suited to deal with highly inhomogeneous one dimensional multifractal measures we find that the fixed mass dual spectrum of generalized dimensions depends on both the system size l and the length n of the sequence considered being stable however when these two parameters are kept fixed a finite size scaling relation is proposed allowing us to define a renormalized spectrum independent of size effects we interpret our results as evidence of extremely long range correlations induced in the sequence by the criticality of the system

multifractal analysis and topological properties of a new family of weighted koch networks abstract weighted complex networks especially scale free networks which characterize real life systems better than non weighted networks have attracted considerable interest in recent years studies on the multifractality of weighted complex networks are still to be undertaken in this paper inspired by the concepts of koch networks and koch island we propose a new family of weighted koch networks and investigate their multifractal behavior and topological properties we find some key topological properties of the new networks their vertex cumulative strength has a power law distribution there is a power law relationship between their topological degree and weight strength the networks have a high weighted clustering coefficient of 0 41004 which is independent of the scaling factor c in the limit of large generation t the second smallest eigenvalue 2 and the maximum eigenvalue n are approximated by quartic polynomials of the scaling factor c for the general laplacian operator while 2 is approximately a quartic polynomial of c and n 1 5 for the normalized laplacian operator then we find that weighted koch networks are both fractal and multifractal their fractal dimension is influenced by the scaling factor c we also apply these analyses to six real world networks and find that the multifractality in three of them are strong

recognition of an organism from fragments of its complete genome this paper considers the problem of matching a fragment to an organism using its complete genome our method is based on the probability measure representation of a genome we first demonstrate that these probability measures can be modeled as recurrent iterated function systems rifs consisting of four contractive similarities our hypothesis is that the multifractal characteristics of the probability measure of a complete genome as captured by the rifs is preserved in its reasonably long fragments we compute the rifs of fragments of various lengths and random starting points and compare with that of the original sequence for recognition using the euclidean distance a demonstration on five randomly selected organisms supports the above hypothesis

correlation property of length sequences based on global structure of the complete genome this paper considers three kinds of length sequences of the complete genome detrended fluctuation analysis spectral analysis and the mean distance spanned within time l are used to discuss the correlation property of these sequences the values of the exponents from these methods of these three kinds of length sequences of bacteria indicate that the long range correlations exist in most of these sequences the correlations have a rich variety of behaviors including the presence of anti correlations furthermore using the exponent gamma it is found that these correlations are all linear gamma 1 0 0 03 it is also found that these sequences exhibit l f noise in some interval of frequency f l the length of this interval of frequency depends on the length of the sequence the shape of the periodogram in f l exhibits some periodicity the period seems to depend on the length and the complexity of the length sequence

multifractal detrended fluctuation analysis based on fractal fitting the long range correlation detection method for highway volume data abstract in this paper we investigate the traffic time series for volume data observed on the guangshen highway we introduce a multifractal detrended fluctuation analysis based on fractal fitting mfdfa ff which is one of the most effective methods to detect long range correlations of time series through effective detecting of long range correlations highway volume can be predicted more accurately in order to get a better detrend effect we use fractal fitting to replace polynomial fitting in detrend process the result shows that fractal fitting can get a better detrend effect than polynomial fitting and the mfdfa ff method can achieve a more accurate research result then we introduce the legendre spectrum to detect the multifractal property characterized by the long range correlation and multifractality of guangshen highway volume data

multifractal analysis of dna sequences using a novel chaos game representation abstract we present a generalization of the standard chaos game representation method introduced by jeffrey to this aim a dna symbolic sequence is mapped onto a singular measure on the attractor of a particular ifs model which is a perfect statistical representation of the sequence a multifractal analysis of the resulting measure is introduced and an interpretation of singularities in terms of mutual information and redundancy statistical dependence among subsequence symbols within the dna sequence is provided the multifractal spectrum is also shown to be more sensitive for detecting dependence structures within the dna sequence than the averaged contribution given by redundancy this method presents several advantages with respect to other representations such as walks or interfaces which may introduce spurious effects in contrast with the results obtained by other standard methods here we note that no general statement can be made on the influence of coding and non coding content on the correlation length of a given sequence

stability analysis and error estimates of local discontinuous galerkin methods with implicit explicit time marching for nonlinear convection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with implicit explicit imex time discretization schemes for solving one dimensional convection diffusion equations with a nonlinear convection both runge kutta and multi step imex methods are considered by the aid of the energy method we show that the imex ldg schemes are unconditionally stable for the nonlinear problems in the sense that the time step is only required to be upper bounded by a positive constant which depends on the flow velocity and the diffusion coefficient but is independent of the mesh size h we also give optimal error estimates for the imex ldg schemes under the same temporal condition if a monotone numerical flux is adopted for the convection numerical experiments are given to verify our main results

multifractality in time series we apply the concepts of multifractal physics to financial time series in order to characterize the onset of crash for the standard poor 500 s p500 stock index x t it is found that within the framework of multifractality the analogous specific heat of the s p500 discrete price index displays a shoulder to the right of the main peak for low time lag values for decreasing t the presence of the shoulder is a consequence of the peaked temporal x t t x t fluctuations in this regime for large time lags t 80 we have found that cq displays typical features of a classical phase transition at a critical point an example of such dynamic phase transition in a simple economic model system based on a mapping with multifractality phenomena in random multiplicative processes is also presented by applying former results obtained with a continuous probability theory for describing scaling measures

multifractal detrended cross correlation analysis of genome sequences using chaos game representation abstract we characterized the multifractal nature and power law cross correlation between any pair of genome sequence through an integrative approach combining 2d multifractal detrended cross correlation analysis and chaos game representation in this paper we have analyzed genomes of some prokaryotes and calculated fractal spectra h q and f from our analysis we observed existence of multifractal nature and power law cross correlation behavior between any pair of genome sequences cluster analysis was performed on the calculated scaling exponents to identify the class affiliation and the same is represented as a dendrogram we suggest this approach may find applications in next generation sequence analysis big data analytics etc

a multifractal formalism let x be a metric space and a borel probability measure on x for q t r and e x write formula then hq t and pq t are borel measures hq t is a multifractal generalization of the centered hausdorff measure and pq t is a multifractal generalization of the packing measure the measures hq t and pq t define for a fixed q in the usual way a generalized hausdorff dimension dimq e and a generalized packing dimension dimq e of subsets e of x we study the functions b q dimq supp b q dimq supp and their relation to the so called multifractal spectra functions of formula we prove among other things that f f is bounded from above by the legendre transform of b b and that equality holds for graph directed self similar measures and cookie cutter measures finally we discuss the connection with generalized renyi dimensions

m form numerical range and the computation of the structured singular value an equivalent expressions for the structures singular value is proposed leading to an alternative algorithm for its computation the approach is based to the geometric properties of the m form numerical range of certain tuples of matrices similar to previously considered schemes the algorithm proposed is proved to give the correct value for block structures of size up to three for larger sizes insight is gained into the question of the possible gap between the structured singular value and its known upper bound

measuring correlations in symbol sequences abstract the paper is devoted to relations between correlation functions and mutual information it is shown that in sequences over an alphabet of symbols statistical dependences are measured by 1 2 independent parameters however not all of them can be determined by autocorrelation functions appropriate sets of correlation functions including crosscorrelations are introduced which allow the detection of all dependences the results are exemplified for binary ternary and quaternary symbol sequences as an application it is discussed that a nonuniform codon usage in protein coding dna sequences introduces periodic correlations even at distances in the order of 1000 base pairs

measuring complexity nonextensivity and chaos in the dna sequence of the major histocompatibility complex abstract we analyze 4 mb sequences of the major histocompatibility complex mhc which is a dna segment on chromosome 6 with high gene density controlling many immunological functions and associated with many diseases the analysis is based on modern theoretical and mathematical tools of complexity theory such as nonlinear time series analysis and tsallis non extensive statistics the results revealed that the dna complexity and self organization can be related to fractional dynamical nonlinear processes with low dimensional deterministic chaotic and non extensive statistical character which generate the dna sequences under the extremization of tsallis q entropy principle while it still remains an open question as to whether the dna walk is a fractional brownian motion fbm a static anomalous diffusion process or a non gaussian dynamical fractional anomalous diffusion process the results of this study testify for the latter providing also a possible explanation for the previously observed long range power law correlations of nucleotides as well as the long range correlation properties of coding and non coding sequences present in dna sequences

runge kutta methods without order reduction for linear initial boundary value problems it is well known the loss of accuracy when a runge kutta method is used together with the method of lines for the full discretization of an initial boundary value problem we show that this phenomenon called order reduction is caused by wrong boundary values in intermediate stages with a right choice the order reduction can be avoided and the optimal order of convergence in time is achieved we prove this fact for time discretizations of abstract initial boundary value problems based on implicit runge kutta methods moreover we apply these results to the full discretization of parabolic problems by means of galerkin finite element techniques we present some numerical examples in order to confirm that the optimal order is actually achieved

factorial moments analyses show a characteristic length scale in dna sequences a unique feature of most of the dna sequences found through the factorial moments analysis is the existence of a characteristic length scale around which the density distribution is nearly poissonian above this point the dna sequences irrespective of their intron contents show long range correlations with a significant deviation from the gaussian statistics while below this point the dna statistics are essentially gaussian the famous dna walk representation is also shown to be a special case of the present analysis

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

magnitude and sign of long range correlated time series decomposition and surrogate signal generation we systematically study the scaling properties of the magnitude and sign of the fluctuations in correlated time series which is a simple and useful approach to distinguish between systems with different dynamical properties but the same linear correlations first we decompose artificial long range power law linearly correlated time series into magnitude and sign series derived from the consecutive increments in the original series and we study their correlation properties we find analytical expressions for the correlation exponent of the sign series as a function of the exponent of the original series such expressions are necessary for modeling surrogate time series with desired scaling properties next we study linear and nonlinear correlation properties of series composed as products of independent magnitude and sign series these surrogate series can be considered as a zero order approximation to the analysis of the coupling of magnitude and sign in real data a problem still open in many fields we find analytical results for the scaling behavior of the composed series as a function of the correlation exponents of the magnitude and sign series used in the composition and we determine the ranges of magnitude and sign correlation exponents leading to either single scaling or to crossover behaviors finally we obtain how the linear and nonlinear properties of the composed series depend on the correlation exponents of their magnitude and sign series based on this information we propose a method to generate surrogate series with controlled correlation exponent and multifractal spectrum

bayesian estimation of self similarity exponent in this study we propose a bayesian approach to the estimation of the hurst exponent in terms of linear mixed models even for unevenly sampled signals and signals with gaps our method is applicable we test our method by using artificial fractional brownian motion of different length and compare it with the detrended fluctuation analysis technique the estimation of the hurst exponent of a rosenblatt process is shown as an example of an h self similar process with non gaussian dimensional distribution additionally we perform an analysis with real data the dow jones industrial average closing values and analyze its temporal variation of the hurst exponent

statistical linguistic characterization of variability in observed and synthetic daily precipitation series abstract this paper deals with an application of zipf law in climatology this analysis allows the extraction of information not available by standard methods in particular rainfall temporal aggregation patterns associated with different climates are characterized by means of exponents derived from the resulting scaling laws the analogy with linguistic analysis is obtained using a particular coding of precipitation as a discrete variable with four states corresponding to four standard precipitation thresholds each weekly symbolic sequence of observed precipitation is considered as a word and each local station defines a language characterized by the observed words in a period representative of the climatology to characterize these precipitation languages we obtained characteristic exponents derived from the zipf law for a set of representative stations of the main koppen s climates and subclimates we found different scaling behaviors for different subclimates given by a single exponent in the range 0 6 humid tropical climates to 1 4 polar climates some humid middle latitude subclimates exhibit a crossover with two different characteristic exponents corresponding to high and low frequency aggregation patterns no explanation for this behavior is provided as an application of the proposed methodology the same analysis was applied to stochastic time series obtained from standard weather generators we found that the language defined by a first order markov process agrees with the observed language of the observed series

analysis of a local discontinuous galerkin method for linear time dependent fourth order problems we analyze a local discontinuous galerkin method for fourth order time dependent problems optimal error estimates are obtained in one dimension and in multidimensions for cartesian and triangular meshes we extend the analysis to higher even order equations and the linearized cahn hilliard type equations numerical experiments are displayed to verify the theoretical results

the correct formulation of intermediate boundary conditions for runge kutta time integration of initial boundary value problems a disturbing feature of applying explicit high order runge kutta rk time integrators to initial boundary value problems solved by the method of lines is the loss of accuracy that results from wrong specifications of intermediate stage boundary conditions the solution proposed here is to prescribe analytically those values that would result from applying the rk solver at the boundaries and hence maintain the order of the rk method without a reduction in the step size limit the procedure is described in detail for nonlinear hyperbolic equations and both scalar and vector examples are presented

dna viewed as an out of equilibrium structure the complexity of the primary structure of human dna is explored using methods from nonequilibrium statistical mechanics dynamical systems theory and information theory a collection of statistical analyses is performed on the dna data and the results are compared with sequences derived from different stochastic processes the use of 2 tests shows that dna can not be described as a low order markov chain of order up to r 6 although detailed balance seems to hold at the level of a binary alphabet it fails when all four base pairs are considered suggesting spatial asymmetry and irreversibility furthermore the block entropy does not increase linearly with the block size reflecting the long range nature of the correlations in the human genomic sequences to probe locally the spatial structure of the chain we study the exit distances from a specific symbol the distribution of recurrence distances and the hurst exponent all of which show power law tails and long range characteristics these results suggest that human dna can be viewed as a nonequilibrium structure maintained in its state through interactions with a constantly changing environment based solely on the exit distance distribution accounting for the nonequilibrium statistics and using the monte carlo rejection sampling method we construct a model dna sequence this method allows us to keep both long and short range statistical characteristics of the native dna data the model sequence presents the same characteristic exponents as the natural dna but fails to capture spatial correlations and point to point details 2014 american physical society

the influence of the finite velocity on spatial distribution of particles in the frame of levy walk model abstract levy walk at the finite velocity is considered to analyze the spatial and temporal characteristics of this process the method of moments has been used the asymptotic distributions of the moments at t have been obtained for n dimensional case where the free path of particles demonstrates the power law distribution p x x 0 x 1 x 0 2 the three regimes of distribution have been distinguished ballistic diffusion and asymptotic introduction of the finite velocity requires considering of two problems propagation with distribution at the finite mathematical expectation of the free path 1 2 and propagation with distribution at the infinite mathematical expectation of the free path of the particle 0 1 in the case 1 2 the asymptotic distribution is described by the levy stable law and the effect of the finite velocity is reduced to a decrease of diffusivity at 0 1 the situation is quite different here the asymptotic distribution exhibits a u or w shape and is described as the ballistic regime of distribution the obtained moments allow to reconstruct the distribution densities of particles in one dimensional and three dimensional cases

fractal analysis of measure representation of large proteins based on the detailed hp model abstract the notion of measure representation of protein sequences is introduced based on the detailed hp model multifractal analysis and detrended fluctuation analysis are then performed on the measure representations of a large number of long protein sequences it is concluded that these protein sequences are not completely random sequences through the measure representations and the values of the d q spectra and related c q curves the values of the exponent from the detrended fluctuation analysis show that the k strings with the ordering in the measure representation exhibit strong long range correlation for substrings with length k 5 the d q spectra of all proteins studied are multifractal like and sufficiently smooth for the c q curves to be meaningful the c q curves of all proteins resemble a classical phase transition at a critical point an ifs model is found to simulate the measure representation of protein sequences very well from the estimated values of parameters in the ifs model we think the non polar residues and uncharged polar residues play a more important role than other kinds of residues in the protein folding process

topological properties and fractal analysis of a recurrence network constructed from fractional brownian motions many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks in this work we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional brownian motions fbms first our results indicate that the constructed recurrence networks have exponential degree distributions the average degree exponent increases first and then decreases with the increase of hurst index h of the associated fbms the relationship between h and can be represented by a cubic polynomial function we next focus on the motif rank distribution of recurrence networks so that we can better understand networks at the local structure level we find the interesting superfamily phenomenon i e the recurrence networks with the same motif rank pattern being grouped into two superfamilies last we numerically analyze the fractal and multifractal properties of recurrence networks we find that the average fractal dimension db of recurrence networks decreases with the hurst index h of the associated fbms and their dependence approximately satisfies the linear formula db 2 h which means that the fractal dimension of the associated recurrence network is close to that of the graph of the fbm moreover our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks and the multifractality of these networks becomes stronger at first and then weaker when the hurst index of the associated time series becomes larger from 0 4 to 0 95 in particular the recurrence network with the hurst index h 0 5 possesses the strongest multifractality in addition the dependence relationships of the average information dimension d 1 and the average correlation dimension d 2 on the hurst index h can also be fitted well with linear functions our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated fbm series

multifractal analysis of nonhyperbolic coupled map lattices application to genomic sequences symbolic sequences generated by coupled map lattices cmls can be used to model the chaotic like structure of genomic sequences in this study it is shown that diffusively coupled chebyshev maps of order 4 corresponding to a shift of four symbols very closely reproduce the multifractal spectrum d q of human genomic sequences for coupling constant 0 35 0 01 if q 0 the presence of rare configurations causes deviations for q 0 which disappear if the rare event statistics of the cml is modified such rare configurations are known to play specific functional roles in genomic sequences serving as promoters or regulatory elements

long range correlation properties of coding and noncoding dna sequences genbank analysis an open question in computational molecular biology is whether long range correlations are present in both coding and noncoding dna or only in the latter to answer this question we consider all 33301 coding and all 29453 noncoding eukaryotic sequences each of length larger than 512 base pairs bp in the present release of the genbank to dtermine whether there is any statistically significant distinction in their long range correlation properties standard fast fourier transform fft analysis indicates that coding sequences have practically no correlations in the range from 10 bp to 100 bp spectral exponent beta 0 00 0 04 where the uncertainty is two standard deviations in contrast for noncoding sequences the average value of the spectral exponent beta is positive 0 16 0 05 which unambiguously shows the presence of long range correlations we also separately analyze the 874 coding and the 1157 noncoding sequences that have more than 4096 bp and find a larger region of power law behavior we calculate the probability that these two data sets coding and noncoding were drawn from the same distribution and we find that it is less than 10 10 we obtain independent confirmation of these findings using the method of detrended fluctuation analysis dfa which is designed to treat sequences with statistical heterogeneity such as dna s known mosaic structure patchiness arising from the nonstationarity of nucleotide concentration the near perfect agreement between the two independent analysis methods fft and dfa increases the confidence in the reliability of our conclusion

convergence and order reduction of runge kutta schemes applied to evolutionary problems in partial differential equations we address the question of convergence of fully discrete runge kutta approximations we prove that under certain conditions the order in time of the fully discrete scheme equals the conventional order of the runge kutta formula being used however these conditions which are necessary for the result to hold are not natural as a result in many problems the order in time will be strictly smaller than the conventional one a phenomenon called order reduction this phenomenon is extensively discussed both analytically and numerically as distinct from earlier contributions we here treat explicit runge kutta schemes although our results are valid for both parabolic and hyperbolic problems the examples we present are therefore taken from the hyperbolic field as it is in this area that explicit discretizations are most appealing

a non extensive approach to the entropy of symbolic sequences symbolic sequences with long range correlations are expected to result in a slow regression to a steady state of entropy increase however we prove that also in this case a fast transition to a constant rate of entropy increase can be obtained provided that the extensive entropy of tsallis with entropic index q is adopted thereby resulting in a new form of entropy that we shall refer to as kolmogorov sinai tsallis kst entropy we assume that the same symbols either 1 or 1 are repeated in strings of length l with the probability distribution p l 1 l the numerical evaluation of the kst entropy suggests that at the value 2 a sort of abrupt transition might occur for the values of in the range 1 3 it is argued that this phase transition like property signals the onset of the thermodynamical regime at 2

error estimates to smooth solutions of runge kutta discontinuous galerkin methods for scalar conservation laws in this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for runge kutta discontinuous galerkin rkdg methods where the time discretization is the second order explicit total variation diminishing tvd runge kutta method error estimates for the mathbb p 1 piecewise linear elements are obtained under the usual cfl condition tau leq gamma h for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions where h and tau are the maximum element lengths and time steps respectively and the positive constant gamma is independent of h and tau however error estimates for higher order mathbb p k k geq 2 elements need a more restrictive time step tau leq gamma h 4 3 we remark that this stronger condition is indeed necessary as the method is linearly unstable under the usual cfl condition tau leq gamma h for the mathbb p k elements of degree k geq 2 error estimates of o h k 1 2 tau 2 are obtained for general monotone numerical fluxes and optimal error estimates of o h k 1 tau 2 are obtained for upwind numerical fluxes

multifractal measures and a weak separation condition abstract we define a new separation property on the family of contractive similitudes that allows certain overlappings this property is weaker than the open set condition of hutchinson it includes the well known class of infinite bernoulli convolutions associated with the p v numbers and the solutions of the two scale dilation equations our main purpose in this paper is to prove the multifractal formalism under such condition

multifractal detrended cross correlation analysis of coding and non coding dna sequences through chaos game representation abstract we propose a new approach combining the chaos game representation and the two dimensional multifractal detrended cross correlation analysis methods to examine multifractal behavior in power law cross correlation between any pair of nucleotide sequences of unequal lengths in this work we analyzed the characteristic behavior of coding and non coding dna sequences of eight prokaryotes the results show the presence of strong multifractal nature between coding and non coding sequences of all data sets we found that this integrative approach helps us to consider complete dna sequences for characterization and further it may be useful for classification clustering identification of class affiliation of nucleotide sequences etc with high precision

on numerical boundary treatment of hyperbolic systems for finite difference and finite element methods it is well known that the stability of the initial boundary value problem for a scalar equation does not necessarily imply stability for a vector equation with a similar boundary treatment in fact we show that for any boundary treatment one can construct systems for which the boundary treatment on the natural variables is not stable our main theorem is that one can introduce a treatment of the boundaries for which the stability of the system follows immediately from the stability for a scalar equation this is accomplished by operating on the characteristic variables for those quantities that are not specified on the boundary in a working code this can be accomplished by adding a correction term to the existing boundary algorithm the analysis and computational results are presented for both finite differences and semi discrete galerkin methods

dynamical approach to levy processes we derive the diffusion process generated by a correlated dichotomous fluctuating variable y starting from a liouville like equation by means of a projection procedure this approach makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable f y t of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d 2 f y t dt 2 furthermore in the special case where f y t has an inverse power law with the index b ranging from 0 to 1 thus making it nonintegrable we show analytically that the

third order implicit explicit runge kutta local discontinuous galerkin methods with suitable boundary treatment for convection diffusion problems with dirichlet boundary conditions abstract to avoid the order reduction when third order implicit explicit runge kutta time discretization is used together with the local discontinuous galerkin ldg spatial discretization for solving convection diffusion problems with time dependent dirichlet boundary conditions we propose a strategy of boundary treatment at each intermediate stage in this paper the proposed strategy can achieve optimal order of accuracy by numerical verification also by suitably setting numerical flux on the boundary in the ldg methods and by establishing an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient and the given boundary conditions we build up the unconditional stability of the corresponding scheme in the sense that the time step is only required to be upper bounded by a suitable positive constant which is independent of the mesh size

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

factorizable language from dynamics to bacterial complete genomes symbolic sequences generated by symbolic dynamics of a dynamical system belong to a special class of language in which any admissible word is factorisable as well as prolongable from a complete genome sequence of an organism one may also define a factorizable language a factorizable language enjoys the nice property that it is entirely determined by the set of minimal fobidden words or distinct excluded blocks debs we use this property to calculate the fractal dimension of patterns related to a visualisation scheme of under represented strings in bacterial complete genomes within the limit of infinitely long strings the same problem may be solved by using a purely combinatorial approach the methods described in this paper may be applied to other regular fractals with self similar and self overlapping structure c 2000 elsevier science b v all rights reserved

thermodynamics of fractal signals based on wavelet analysis application to fully developed turbulence data and dna sequences we use the continuous wavelet transform to generalize the multifractal formalism to fractal functions we report the results of recent applications of the so called wavelet transform modulus maxima wtmm method to fully developed turbulence data and dna sequences we conclude by briey describing some works currently under progress which are likely to be the guidelines for future research c 1998 elsevier science b v all rights reserved

measure representation and multifractal analysis of complete genomes this paper introduces the notion of measure representation of dna sequences spectral analysis and multifractal analysis are then performed on the measure representations of a large number of complete genomes the main aim of this paper is to discuss the multifractal property of the measure representation and the classification of bacteria from the measure representations and the values of the dq spectra and related cq curves it is concluded that these complete genomes are not random sequences in fact spectral analyses performed indicate that these measure representations considered as time series exhibit strong long range correlation here the long range correlation is for the k strings with dictionary ordering and it is different from the base pair correlations introduced by other people for substrings with length k58 the dq spectra of all organisms studied are multifractal like and sufficiently smooth for the cq curves to be meaningful with the decreasing value of k the multifractality lessens the cq curves of all bacteria resemble a classical phase transition at a critical point but the analogous phase transitions of chromosomes of nonbacteria organisms are different apart from chromosome 1 of c elegans they exhibit the shape of double peaked specific heat function a classification of genomes of bacteria by assigning to each sequence a point in two dimensional space d21 d1 and in three dimensional space d21 d1 d22 was given bacteria that are close phylogenetically are almost close in the spaces d21 d1 and d21 d1 d22

λ matrix formulation applied to the hertz contact problem with finite friction in this paper we are concerned with coupled volterra integral equations for rigid bodies in the frame of the hertz contact model considering an appropriate integral operator as the variable of the matrix polynomial a l matrix corresponding to the hertz model is determined knowledge of the smith decomposition forms of the l matrices enables us to characterize a single integral equation which recovers the normal stress as well as the compliance factor the presented model shows a reliable behavior and an acceptable accuracy of the l matrix approach in comparison to the recent results in the literature

multifractal characterisation of length sequences of coding and noncoding segments in a complete genome the coding and noncoding length sequences constructed from a complete genome are characterised by multifractal analysis the dimension spectrum dq and its derivative the analogous specific heat cq are calculated for the coding and noncoding length sequences of bacteria where q is the moment order of the partition sum of the sequences from the shape of the dq and cq curves it is seen that there exists a clear difference between the coding noncoding length sequences of all organisms considered and a completely random sequence the complexity of noncoding length sequences is higher than that of coding length sequences for bacteria almost all dq curves for coding length sequences are flat so their multifractality is small whereas almost all dq curves for noncoding length sequences are multifractal like it is seen that the analogous specific heats of noncoding length sequences of bacteria have a rich variety of behaviour which is much more complex than that of coding length sequences we propose to characterise the bacteria according to the types of the cq curves of their noncoding length sequences this new type of classification allows a better understanding of the relationship among bacteria at the global gene level instead of nucleotide sequence level

statistics of dna sequences a low frequency analysis we study statistical properties of dna chains of thirteen microbial complete genomes we find that the power spectrum of several of the sequences studied flattens off in the low frequency limit this implies the correlation length in those sequences is much smaller than the entire dna chain consequently in contradiction with previous studies we show that the fractal behavior of dna chains does not always prevail through the entire dna molecule

on the removal of boundary errors caused by runge kutta integration of non linear partial differential equations the temporal integration of hyperbolic partial differential equations pdes has been shown to lead sometimes to the deterioration of accuracy of the solution because of boundary conditions a procedure for removal of this error in the linear case has been established previously in this paper we consider hyperbolic pdes linear and nonlinear whose boundary treatment is accomplished via the simultaneous approximation term sat procedure a methodology is presented for recovery of the full order of accuracy and has been applied to the case of a fourth order explicit finite difference scheme

what can we learn with wavelets about dna sequences we use the wavelet transform to explore the complexity of dna sequences long range correlations are clearly identified and shown to be related to the sequence gc content the significance of this observation to gene evolution is discussed

symbolic strings and spatial 1 f spectra abstract some string dynamics which are expected to show non trivial correlations are proposed and studied it is because we were motivated by the observation that some dna sequences show spatial 1 f correlation our model is based on a rule by which one string is split into two parts in the middle and another string is inserted between them when the strings are made longer the basic model shows long range correlations with a structure characteristic of cyclic patterns on the log log plot of the spectrum the effects of string cross over and point mutation are also studied the cross over destroys the structures appearing in the basic model and makes 1 f like correlation clear the point mutation makes the power spectrum flat as is expected from the white noise nature the analogy of the cross over model with real dna evolution dynamics is suggested

imex runge kutta schemes for reaction diffusion equations a fundamental research is carried out into convergence and stability properties of imex implicit explicit runge kutta schemes applied to reaction diffusion equations it is shown that a fully discrete scheme converges if it satisfies certain conditions using a technique of the b convergence analysis developed by burrage hundsdorfer and verwer in 1986 stability of the schemes is also examined on the basis of a scalar test equation proposed by frank hundsdorfer and verwer in 1997

the theoretical accuracy of runge kutta time discretizations for the initial boundary value problem a study of the boundary error the conventional method of imposing time dependent boundary conditions for runge kutta rk time advancement reduces the formal accuracy of the space time method to first order locally and second order globally independently of the spatial operator this counter intuitive result is analyzed in this paper two methods of eliminating this problem are proposed for the linear constant coefficient case 1 impose the exact boundary condition only at the end of the complete rk cycle 2 impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives the first method while retaining the rk accuracy in all cases results in a scheme with much reduced cfl condition rendering the rk scheme less attractive the second method retains the same allowable time step as the periodic problem however it is a general remedy only for the linear case for non linear hyperbolic equations the second method is effective only for for rk schemes of third order accuracy or less numerical studies are presented to verify the efficacy of each approach

the eigenstructure of an arbitrary polynomial matrix computational aspects we give a new numerical method to compute the eigenstructure i e the zero structure the polar structure and the left and right null space structure of a polynomial matrix p these structural elements are of fundamental importance in several systems and control problems involving polynomial matrices the approach is more general than previous numerical methods because it can be applied to an arbitrary m n polynomial matrix p with normal rank r smaller than m and or n the algorithm is then shown to compute the structure of the left and right null spaces of p as well the speed and accuracy of this new approach are also discussed

estimating the intermittency of point processes with applications to human activity and viral dna abstract the intermittency of a point process is the extent to which the number of events in a time window has pronounced departures from typical values combining point process and multifractal formalisms indicates that the correlation codimension can be used to quantify intermittency the correlation codimension is easily estimated and is simply related to other second order scaling exponents such as those of the fano factor and spectral density the correlation codimensions are derived for various uncorrelated fractal and fractal rate point processes in addition the estimation of intermittency as the correlation codimension of experimental events is illustrated with applications to experimental data human activity during bed rest is highly intermittent while other human activity and viral dna composition are non intermittent

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

a study of the time distribution of inter cluster earthquakes in taiwan abstract in this study we analyze the time distribution of inter cluster earthquakes in taiwan to infer correlation of earthquake occurrences our study based on the diffusion entropy analysis dea shows that there is positive correlation when the event time series is sampled by a time unit ranging from 120 to 1800 min also in the range of the time unit two slope feature in the determination of scaling exponent in dea is observed and the scenario is demonstrated by the simulations of synthetic earthquake sequence generated by the copying mistake map model our simulation suggests that the earthquake sequence may consist of a randomness sequence and a correlated sequence and the latter contributes to the correlation observed in our analysis