fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

order preserving pattern matching with k mismatches we consider a generalization of the recently introduced order preserving pattern matching the difference between the standard pattern matching and the order preserving variant is that instead of looking for an exact copy of the pattern we only require that the relative order between the elements is the same in our generalization we additionally allow up to k mismatches between the pattern of length m and the text of length n and the goal is to construct an efficient algorithm for small values of k our solution detects an order preserving occurrence with up to k mismatches in o n loglogm kloglogk time

existence of continuous solutions and stability for a general iterative functional equation in this paper c 0 solutions for a general form of iterative functional equation are considered when the coefficient of the first order iterate can be zero which is called the leading coefficient problem for polynomial like iterative equations moreover the stability of those c 0 solutions is also discussed finally as an application we give an algorithm to compute the piecewise linear solutions of polynomial like iterative equations

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

subdiffusive master equation with space dependent anomalous exponent and structural instability we derive the fractional master equation with space dependent anomalous exponent we analyze the asymptotic behavior of the corresponding lattice model both analytically and by monte carlo simulation we show that the subdiffusive fractional equations with constant anomalous exponent in a bounded domain 0 l are not structurally stable with respect to the nonhomogeneous variations of parameter in particular the gibbs boltzmann distribution is no longer the stationary solution of the fractional fokker planck equation whatever the space variation of the exponent might be we analyze the random distribution of in space and find that in the long time limit the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events we show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a black swan the low probability event of the small value of the anomalous exponent that completely dominates the long time behavior of subdiffusive systems

linear response in complex systems ctrw and the fractional fokker planck equations abstract we consider the linear response of systems modelled by continuous time random walks ctrw and by fractional fokker planck equations under the influence of time dependent external fields we calculate the corresponding response functions explicitly the ctrw curve exhibits aging i e it is not translationally invariant in the time domain this is different from what happens under fractional fokker planck conditions

a really simple approximation of smallest grammar in this paper we present a really simple linear time algorithm constructing a context free grammar of size 4 g log 3 2 n g for the input string where n is the size of the input string and g the size of the optimal grammar generating this string the algorithm works for arbitrary size alphabets but the running time is linear assuming that the alphabet of the input string can be identified with numbers from 1 n c for some constant c algorithms with such an approximation guarantee and running time are known however all of them were non trivial and their analyses were involved the here presented algorithm computes the lz77 factorisation and transforms it in phases to a grammar in each phase it maintains an lz77 like factorisation of the word with at most factors as well as additional o letters where was the size of the original lz77 factorisation in one phase in a greedy way by a left to right sweep and a help of the factorisation we choose a set of pairs of consecutive letters to be replaced with new symbols i e nonterminals of the constructed grammar we choose at least 2 3 of the letters in the word and there are o many different pairs among them hence there are o log n phases each of them introduces o nonterminals to a grammar a more precise analysis yields a bound 4 log n as g this yields the desired bound g 4 g log n g

computer simulation of geometric stable distributions we present a new method for computer simulation of strictly geometric stable random variables the method is based on their representation as a product of two independent random variables with explicit distribution functions coupled with the inversion method we also extend the method to the multivariate case by deriving new representation and simulation algorithm for multivariate linnik distribution

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

a survey of string orderings and their application to the burrows wheeler transform abstract for over 20 years the data clustering properties and applications of the efficient burrows wheeler transform have been researched lexicographic suffix sorting is induced during the transformation and more recently a new direction has considered alternative ordering strategies for suffix arrays and thus the transforms in this survey we look at these distinctly ordered bijective and linear transforms for arbitrary alphabets we discuss the v bwt derived from v order and the d bwt based on lex extension order the binary case yields a pair of transforms the binary rouen b bwt defined using binary block order lyndon words are relevant to implementing the original transform the new transforms are defined for analogous structures v words indeterminate lyndon words and b words respectively there is plenty of scope for further non lexicographic transforms as indicated in the conclusion

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

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

stochastic solution of space time fractional diffusion equations classical and anomalous diffusion equations employ integer derivatives fractional derivatives and other pseudodifferential operators in space in this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are mittag leffler type this leads to explicit solutions for space time fractional diffusion equations with multiscaling space fractional derivatives and additional insight into the meaning of these equations

faster suffix sorting we propose a fast and memory efficient algorithm for lexicographically sorting the suffixes of a string a problem that has important applications in data compression as well as string matching our algorithm eliminates much of the overhead of previous specialized approaches while maintaining their robustness for degenerate inputs for input size n our algorithm operates in only two integer arrays of size n and has worst case time complexity o nlogn we demonstrate experimentally that our algorithm has stable performance compared with other approaches

efficient computation of longest single arm gapped palindromes in a string abstract in this paper we introduce new types of approximate palindromes called single arm gapped palindromes shortly sagps a sagp contains a gap in either its left or right arm which is in the form of either w g u c u r w r or w u c u r g w r where w and u are non empty strings w r and u r are respectively the reversed strings of w and u g is a string called a gap and c is either a single character or the empty string here we call wu and u r w r the arm of the sagp and u v the length of the arm we classify sagps into two groups those which have u c u r as a maximal palindrome type 1 and the others type 2 we propose several algorithms to compute type 1 sagps with longest arms occurring in a given string based on suffix arrays then we propose a linear time algorithm to compute all type 1 sagps with longest arms based on suffix trees also we show how to compute type 2 sagps with longest arms in linear time we also perform some preliminary experiments to show practical performances of the proposed methods

characterizations and simulations of a class of stochastic processes to model anomalous diffusion in this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusions this class called generalized grey brownian motion ggbm is made up of self similar with stationary increments processes h sssi and depends on two real parameters 0 2 and 0 1 it includes fractional brownian motion when 0 2 and 1 and time fractional diffusion stochastic processes when 0 1 the latter have a marginal probability density function governed by timefractional diffusion equations of order the ggbm is defined through the explicit construction of the underlying probability space however in this paper we show that it is possible to define it in an unspecified probability space for this purpose we write down explicitly all the finite dimensional probability density functions moreover we provide different ggbm characterizations the role of them wright function which is related to the fundamental solution of the time fractional diffusion equation emerges as a natural generalization of the gaussian distribution furthermore we show that the ggbm can be represented in terms of the product of a random variable which is related to the m wright function and an independent fractional brownian motion this representation highlights the h sssi nature of the ggbm and provides a way to study and simulate the trajectories for this purpose we developed a random walk model based on a finite difference approximation of a partial integro differential equation of a fractional type

fractional fokker planck dynamics numerical algorithm and simulations anomalous transport in a tilted periodic potential is investigated numerically within the framework of the fractional fokker planck dynamics via the underlying continuous time random walk an efficient numerical algorithm is developed which is applicable for an arbitrary potential this algorithm is then applied to investigate the fractional current and the corresponding nonlinear mobility in different washboard potentials normal and fractional diffusion are compared through their time evolution of the probability density in state space moreover we discuss the stationary probability density of the fractional current values

polymer translocation through a nanopore a showcase of anomalous diffusion the translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments fractional calculus and computer simulations the problem at hand is mapped on a one dimensional anomalous diffusion process in terms of the reaction coordinate s i e the translocated number of segments at time t and shown to be governed by a universal exponent ensuremath alpha 2 2 ensuremath nu 2 ensuremath ensuremath gamma 1 where ensuremath nu is the flory exponent and ensuremath gamma 1 is the surface exponent remarkably it turns out that the value of ensuremath alpha is nearly the same in two and three dimensions the process is described by a fractional diffusion equation which is solved exactly in the interval 0lsln with appropriate boundary and initial conditions the solution gives the probability distribution of translocation times as well as the variation with time of the statistical moments s t and s 2 t ensuremath s t 2 which provide a full description of the diffusion process the comparison of the analytic results with data derived from extensive monte carlo simulations reveals very good agreement and proves that the diffusion dynamics of unbiased translocation through a nanopore is anomalous in its nature

variational principle for fractional kinetics and the levy ansatz a variational principle is developed for fractional kinetics based on the auxiliary field formalism it is applied to the fokker planck equation with spatiotemporal fractionality and a variational solution is obtained with the help of the levy ansatz it is shown how the whole range from subdiffusion to superdiffusion is realized by the variational solution as a competing effect between the long waiting time and the long jump the motion of the center of the probability distribution is also analyzed in the case of a periodic drift

binary block order rouen transform novel twin binary burrows wheeler type transforms are introduced the transforms are defined for lyndon like b words which apply binary block order we call this approach the b bwt rouen transform these bijective rouen transforms and inverses are computed in linear time preliminary experimental results indicate potential value of binary transforms we introduce bijective burrows wheeler type transforms for binary strings 1 the original method by burrows and wheeler 4 is based on lexicographic order for general alphabets and the transform is defined to be the last column of the ordered bwt matrix this new approach applies binary block order b order which yields not one but twin transforms one based on lyndon words the other on a repetition of lyndon words these binary b bwt transforms are constructed here for b words analogous structures to lyndon words a key computation in the transforms is the application of a linear time suffix sorting technique such as 18 21 22 27 to sort the cyclic rotations of a binary input string into their b order moreover like the original lexicographic transform we show that computing the b bwt inverses is also achieved in linear time by using straightforward combinatorial arguments

continuous time random walk galilei invariance and relation for the nth moment we consider a decoupled continuous time random walk model with a generic waiting time probability density function pdf for the force free case we derive an integro differential diffusion equation which is related to the galilei invariance for the probability density we also derive a general relation which connects the nth moment in the presence of any external force to the second moment without external force i e it is valid for any waiting time pdf this general relation includes the generalized second einstein relation which connects the first moment in the presence of any external force to the second moment without any external force these expressions for the first two moments are verified by using several kinds of the waiting time pdf moreover we present new anomalous diffusion behaviours for a waiting time pdf given by a product of power law and exponential function

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

a random telegraph signal of mittag leffler type a general method is presented to explicitly compute autocovariance functions for non poisson dichotomous noise based on renewal theory the method is specialized to a random telegraph signal of mittag leffler type analytical predictions are compared to monte carlo simulations non poisson dichotomous noise is non stationary and standard spectral methods fail to describe it properly as they assume stationarity

on the sorting complexity of suffix tree construction the suffix tree of a string is the fundamental data structure of combinatorial pattern matching we present a recursive technique for building suffix trees that yields optimal algorithms in different computational models sorting is an inherent bottleneck in building suffix trees and our algorithms match the sorting lower bound specifically we present the following results 1 weiner 1973 who introduced the data structure gave an optimal 0 n time algorithm for building the suffix tree of an n character string drawn from a constant size alphabet in the comparison model there is a trivial o n log n time lower bound based on sorting and weiner s algorithm matches this bound for integer alphabets the fastest known algorithm is the o n log n time comparison based algorithm but no super linear lower bound is known closing this gap is the main open question in stringology we settle this open problem by giving a linear time reduction to sorting for building suffix trees since sorting is a lower bound for building suffix trees this algorithm is time optimal in every alphabet mode in particular for an alphabet consisting of integers in a polynomial range we get the first known linear time algorithm 2 all previously known algorithms for building suffix trees exhibit a marked absence of locality of reference and thus they tend to elicit many page faults i os when indexing very long strings they are therefore unsuitable for building suffix trees in secondary storage devices where i os dominate the overall computational cost we give a linear i o reduction to sorting for suffix tree construction since sorting is a trivial i o lower bound for building suffix trees our algorithm is i o optimal

a bijective variant of the burrows wheeler transform using v order in this paper we introduce the v transform v bwt a variant of the classic burrows wheeler transform the original bwt uses lexicographic order whereas we apply a distinct total ordering of strings called v order v order string comparison and lyndon like factorization of a string x x 1 n x x 1 n into v words have recently been shown to be linear in their use of time and space daykin et al 2011 18 here we apply these subcomputations along with n n suffix sorting ko and aluru 2003 26 to implement linear v sorting of all the rotations of a string when it is known that the input string x is a v word we compute the v transform in n n time and space and also outline an efficient algorithm for inverting the v transform and recovering x we further outline a bijective algorithm in the case that x is arbitrary we propose future research into other variants of transforms using lex extension orderings daykin et al 2013 19 motivation for this work arises in possible applications to data compression

fast bwt in small space by blockwise suffix sorting abstract we present a new space and time efficient algorithm for computing the burrow wheeler transform bwt for any choice of a parameter v 3 n 2 3 the computation of bwt for a text of length n takes o n log n v n worst case time and o n log n v n average case time using o n log n v bits of space in addition to the text and the bwt for example if v log 2 n the time is o n log 2 n in the worst case and o n log n on an average with the additional space requirement of o n bits the algorithm is alphabet independent it uses only character comparisons and the complexities do not depend on the alphabet size unless v does a practical implementation is 2 3 times slower than one of the fastest and most space efficient previous algorithms while needing only one third of the main memory the algorithm is based on suffix arrays but unlike any other algorithm it can construct the suffix array a small block at a time without storing the rest of the suffix array anywhere

suffix arrays a new method for on line string searches a new and conceptually simple data structure called a suffix array for on line string searches is introduced in this paper constructing and querying suffix arrays is reduced to a sort and search paradigm that employs novel algorithms the main advantage of suffix arrays over suffix trees is that in practice they use three to five times less space from a complexity standpoint suffix arrays permit on line string searches of the type is w a substring of a to be answered in time o p log n where p is the length of w and n is the length of a which is competitive with and in some cases slightly better than suffix trees the only drawback is that in those instances where the underlying alphabet is finite and small suffix trees can be constructed in o n time in the worst case versus o n log n time for suffix arrays however an augmented algorithm is given that regardless of the alphabet size constructs suffix arrays in o n expected time albeit with lesser space efficiency it is

fully functional suffix trees and optimal text searching in bwt runs bounded space indexing highly repetitive texts such as genomic databases software repositories and versioned text collections has become an important problem since the turn of the millennium a relevant compressibility measure for repetitive texts is r the number of runs in their burrows wheeler transforms bwts one of the earliest indexes for repetitive collections the run length fm index used o r space and was able to efficiently count the number of occurrences of a pattern of length m in a text of length n in o m log log n time with current techniques however it was unable to locate the positions of those occurrences efficiently within a space bounded in terms of r in this article we close this long standing problem showing how to extend the run length fm index so that it can locate the occ occurrences efficiently in o occ log log n time within o r space by raising the space to o r log log n our index counts the occurrences in optimal time o m and locates them in optimal time as well o m occ by further raising the space by an o w log factor where is the alphabet size and w e log n is the ram machine size in bits we support count and locate in o m log w and o m log w occ time which is optimal in the packed setting and had not been obtained before in compressed space we also describe a structure using o r log n r space that replaces the text and extracts any text substring of length e in the almost optimal time o log n r e log w within that space we similarly provide access to arbitrary suffix array inverse suffix array and longest common prefix array cells in time o log n r and extend these capabilities to full suffix tree functionality typically in o log n r time per operation our experiments show that our o r space index outperforms the space competitive alternatives by 1 2 orders of magnitude in time competitive implementations of the original fm index are outperformed by 1 2 orders of magnitude in space and or 2 3 in time

fluid limit of the continuous time random walk with general levy jump distribution functions the continuous time random walk ctrw is a natural generalization of the brownian random walk that allows the incorporation of waiting time distributions psi t and general jump distribution functions eta x there are two well known fluid limits of this model in the uncoupled case for exponential decaying waiting times and gaussian jump distribution functions the fluid limit leads to the diffusion equation on the other hand for algebraic decaying waiting times psi approximately t 1 beta and algebraic decaying jump distributions eta approximately x 1 alpha corresponding to levy stable processes the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time however these are two special cases of a wider class of models here we consider the ctrw for the most general levy stochastic processes in the levy khintchine representation for the jump distribution function and obtain an integrodifferential equation describing the dynamics in the fluid limit the resulting equation contains as special cases the regular and the fractional diffusion equations as an application we consider the case of ctrws with exponentially truncated levy jump distribution functions in this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory long jumps and truncation effects in the intermediate asymptotic regime the dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as tauc approximately lambda alpha beta where 1 lambda is the truncation length scale the asymptotic behavior of the propagator green s function of the truncated fractional equation exhibits a transition from algebraic decay for t tauc

radiative transport equation for the mittag leffler path length distribution in this paper we consider the radiative transport equation for infinitely extended scattering media that are characterized by the mittag leffler path length distribution p l le tl which is a generalization of the usually assumed lambert beer law p l texp tl in this context we derive the infinite space green s function of the underlying fractional transport equation for the spherically symmetric medium as well as for the one dimensional string moreover simple analytical solutions are presented for the prediction of the radiation field in the single scattering approximation the resulting equations are compared with monte carlo simulations in the steady state and time domain showing within the stochastic nature of the simulations an excellent agreement

linear work suffix array construction suffix trees and suffix arrays are widely used and largely interchangeable index structures on strings and sequences practitioners prefer suffix arrays due to their simplicity and space efficiency while theoreticians use suffix trees due to linear time construction algorithms and more explicit structure we narrow this gap between theory and practice with a simple linear time construction algorithm for suffix arrays the simplicity is demonstrated with a c implementation of 50 effective lines of code the algorithm is called dc3 which stems from the central underlying concept of difference cover this view leads to a generalized algorithm dc that allows a space efficient implementation and moreover supports the choice of a space time tradeoff for any v 1 n it runs in o vn time using o n v space in addition to the input string and the suffix array we also present variants of the algorithm for several parallel and hierarchical memory models of computation the algorithms for bsp and erew pram models are asymptotically faster than all previous suffix tree or array construction algorithms

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

on a connection between a class of q deformed algebras and the hausdorff derivative in a medium with fractal metric abstract over the recent decades diverse formalisms have emerged that are adopted to approach complex systems amongst those we may quote the q calculus in tsallis version of non extensive statistics with its undeniable success whenever applied to a wide class of different systems kaniadakis approach based on the compatibility between relativity and thermodynamics fractional calculus fc that deals with the dynamics of anomalous transport and other natural phenomena and also some local versions of fc that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations the question we might ask is whether or not there are common aspects that connect these alternative approaches in this short communication we discuss a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely tsallis framework and kaniadakis scenario with local form of fractional derivative operators defined in fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure this connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics such as porous or granular media possible connections with other alternative definitions of fc are also contemplated insights on complexity connected to concepts like coarse grained space time and physics in general are pointed out

alphabet independent linear time construction of compressed suffix arrays using o n log n bit working space the suffix array is a fundamental index data structure in string algorithms and bioinformatics and the compressed suffix array csa and the fm index are its compressed versions many algorithms for constructing these index data structures have been developed recently hon et al w k hon k sadakane w k sung breaking a time and space barrier in constructing full text indices in proceedings of the 44th annual ieee symposium on foundations of computer science 2003 pp 251 260 proposed a construction algorithm using o nloglog time and o nlog bit working space which is the fastest algorithm using o nlog bit working space in this paper we give an efficient algorithm to construct the index data structures our algorithm constructs the suffix array the csa the fm index and burrows wheeler transform using alphabet independent o n time and bit working space where log32 our algorithm takes less time and more space than hon et al s algorithm our algorithm uses least working space among alphabet independent linear time algorithms

linear time computation of prefix table for weighted strings applications the prefix table of a string is one of the most fundamental data structures of algorithms on strings it determines the longest factor at each position of the string that matches a prefix of the string it can be computed in time linear with respect to the size of the string and hence it can be used efficiently for locating patterns or for regularity searching in strings a weighted string is a string in which a set of letters may occur at each position with respective occurrence probabilities weighted strings also known as position weight matrices or uncertain strings naturally arise in many biological contexts for example they provide a method to realise approximation among occurrences of the same dna segment in this article given a weighted string x of length n and a constant cumulative weight threshold 1 z defined as the minimal probability of occurrence of factors in x we present an o n time algorithm for computing the prefix table of x furthermore we outline a number of applications of this result for solving various problems on non standard strings and present some preliminary experimental results

random walk approximation of fractional order multiscaling anomalous diffusion random walks are developed to approximate the solutions of multiscaling fractional order anomalous diffusion equations the essential elements of the diffusion are described by the matrix order scaling indexes and the mixing measure which describes the diffusion coefficient in every direction two forms of the governing equation also called the multiscaling fractional diffusion equation based on fractional flux and fractional divergence are considered where the diffusion coefficient and the drift vary in space the particle tracking algorithm is also extended to approximate anomalous diffusion with a streamline dependent mixing measure using a streamline projection technique in this and other general cases the random walk method is the only known way to solve the nonhomogeneous equations five numerical examples demonstrate the flexibility simplicity and efficiency of the random walk method

parallel computation of the burrows wheeler transform in compact space abstract the burrows wheeler transform bwt has become since its introduction a key tool for representing large text collections in compressed space while supporting indexed searching on a text of length n over an alphabet of size it requires o n lg bits of space instead of the o n lg n bits required by classical indexes a challenge for its adoption is to build it within o n lg bits as well there are some sequential algorithms building it within this space but no such a parallel algorithm in this paper we present a pram crew algorithm to build the bwt using o n lg n work o lg 3 n lg depth and o n lg bits

optimal and sublogarithmic time randomized parallel sorting algorithms this paper assumes a parallel ram random access machine model which allows both concurrent reads and concurrent writes of a global memory the main result is an optimal randomized parallel algorithm for integer sort i e for sorting n integers in the range 1 n this algorithm costs only logarithmic time and is the first known that is optimal the product of its time and processor bounds is upper bounded by a linear function of the input size also given is a deterministic sublogarithmic time algorithm for prefix sum in addition this paper presents a sublogarithmic time algorithm for obtaining a random permutation of n elements in parallel and finally sublogarithmic time algorithms for general sort and integer sort are presented our sub logarithmic general sort algorithm is also optimal

the pseudopalindromic completion of regular languages pseudopalindromes are words that are fixed points for some antimorphic involution in this paper we discuss a newer word operation that of pseudopalindromic completion in which symbols are added to either side of the word such that the new obtained words are pseudopalindromes this notion represents a particular type of hairpin completion where the length of the hairpin is at most one we give precise descriptions of regular languages that are closed under this operation and show that the regularity of the closure under the operation is decidable

a quick tour on suffix arrays and compressed suffix arrays suffix arrays are a key data structure for solving a run of problems on texts and sequences from data compression and information retrieval to biological sequence analysis and pattern discovery in their simplest version they can just be seen as a permutation of the elements in 1 2 n encoding the sorted sequence of suffixes from a given text of length n under the lexicographic order yet they are on a par with ubiquitous and sophisticated suffix trees over the years many interesting combinatorial properties have been devised for this special class of permutations for instance they can implicitly encode extra information and they are a well characterized subset of the n permutations this paper gives a short tutorial on suffix arrays and their compressed version to explore and review some of their algorithmic features discussing the space issues related to their usage in text indexing combinatorial pattern matching and data compression

compressed property suffix trees property matching is a biologically motivated problem where the task is to find those occurrences of an online pattern p in a string text t of size n such that the matched text part satisfies some conceptual property the property of a string is a set p of possibly overlapping intervals s 1 f 1 s 2 f 2 corresponding to the part of text and an occurrence of a pattern p t i i p 1 is a valid output only if t i i p 1 is completely contained in at least one interval s j f j email protected the indexing version of this problem was introduced by a amir 2008 where the text is preprocessed in o email protected nloglogn time and an o nlogn bits index named property suffix tree pst is maintained pst can perform property matching in o p email protected occ p time where occ p is the number of occurrences of p in t satisfying the property t kopelowitz 2010 considered the dynamic version of this problem where intervals can be added or deleted however all these indexes take space linear to the size of text o nlogn bits which can be much more than the size of the text email protected bits in this paper we propose the first index for property matching occupying space close to the entropy compressed space requirement of the text our compressed index takes csa n email protected o 1 bits space and performs query answering in o t p email protected 1 occ p t s a time where csa is the size of compressed suffix array of t t p be the time for searching a pattern of length p in csa t s a is the time for computing the suffix array value and e 0 is a constant we also introduce a dynamic index which takes csa o n p logn bits space and performs query answering in o t p 1 occ p logn t s a logn loglogn time and can update insert delete an interval s f in o f s logn t s a time

short note on the emergence of fractional kinetics abstract in the present short note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non markovian master equations particle trajectories are assumed to be solely markovian and described by the continuous time random walk model but as a consequence of the complexity of the medium each trajectory is supposed to scale in time according to a particular random timescale the link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed in particular when a stationary distribution is considered the timescale distribution is uniquely determined as a function related to the fundamental solution of the space time fractional diffusion equation in contrast when the non stationary case is considered the timescale distribution is no longer unique two distributions are here computed one related to the m wright mainardi function which is green s function of the time fractional diffusion equation and another related to the mittag leffler function which is the solution of the fractional relaxation equation

monte carlo simulation of uncoupled continuous time random walks yielding a stochastic solution of the space time fractional diffusion equation we present a numerical method for the monte carlo simulation of uncoupled continuous time random walks with a levy stable distribution of jumps in space and a mittag leffler distribution of waiting times and apply it to the stochastic solution of the cauchy problem for a partial differential equation with fractional derivatives both in space and in time the one parameter mittag leffler function is the natural survival probability leading to time fractional diffusion equations transformation methods for mittag leffler random variables were found later than the well known transformation method by chambers mallows and stuck for levy stable random variables and so far have not received as much attention nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the mittag leffler distribution combining the two methods we obtain an accurate approximation of space and time fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

searching for gapped palindromes we study the problem of finding in a given word all maximal gapped palindromes verifying two types of constraints that we call long armed and length constrained palindromes for each of the two classes we propose an algorithm that runs in time o n s for a constant size alphabet where s is the number of output palindromes both algorithms can be extended to compute biological gapped palindromes within the same time bound

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena

connecting complexity with spectral entropy using the laplace transformed solution to the fractional diffusion equation analytical solutions to the fractional diffusion equation are often obtained by using laplace and fourier transforms which conveniently encode the order of the time and the space derivatives and as non integer powers of the conjugate transform variables s and k for the spectral and the spatial frequencies respectively this study presents a new solution to the fractional diffusion equation obtained using the laplace transform and expressed as a fox s h function this result clearly illustrates the kinetics of the underlying stochastic process in terms of the laplace spectral frequency and entropy the spectral entropy is numerically calculated by using the direct integration method and the adaptive gauss kronrod quadrature algorithm here the properties of spectral entropy are investigated for the cases of sub diffusion and super diffusion we find that the overall spectral entropy decreases with the increasing and and that the normal or gaussian case with 1 and 2 has the lowest spectral entropy i e less information is needed to describe the state of a gaussian process in addition as the neighborhood over which the entropy is calculated increases the spectral entropy decreases which implies a spatial averaging or coarse graining of the material properties consequently the spectral entropy is shown to provide a new way to characterize the temporal correlation of anomalous diffusion future studies should be designed to examine changes of spectral entropy in physical chemical and biological systems undergoing phase changes chemical reactions and tissue regeneration