generalized chapman kolmogorov equation a unifying approach to the description of anomalous transport in external fields the generalized chapman kolmogorov equation v m kenkre e w montroll and m f shlesinger j stat phys 9 45 1973 is discussed it is demonstrated that this equation unifies recently proposed kinetic equations of fractional order that describe anomalous transport in external fields as well as continuous time random walks the conditions under which the individual models can be established are discussed

absorbing boundary in one dimensional anomalous transport in this paper we study the space time probability distribution q x t of a random walk subject to an absorbing boundary at the origin x 0 for motion controlled by l evy flights and l evy walks characterized by the exponent ensuremath gamma we find that the method of images usually applicable to brownian motion may break down for l evy processes we calculate the distribution q x t to be at xg0 after time tg0 having started at the origin assuming that the boundary is effective at time tg0 we show that q x t depends on the details of the underlying process q x t ensuremath sim mathit x ensuremath gamma 2 mathit t 1 1 ensuremath gamma 1 ensuremath le ensuremath gamma ensuremath le 2 for small x while total survival is independent of the spatial realization of motion and displays a universal behavior we also discuss the related smoluchowski boundary condition problem

an extension of the dynamical foundation for the statistical equilibrium concept abstract this paper reviews a recently introduced generalization of dynamical stationarity involving the appearance of stable convolution semigroups in the ultralong time limit dynamical stationarity is the basis of the equilibrium concept in statistical mechanics and the ultralong time limit is a limit in which a discretized time flow is iterated infinitely often while the discretization time step becomes infinite the new limit is necessary when investigating induced automorphisms for subsets of measure zero it is found that the induced dynamics of subsets of zero measure is given generically by stable convolution semigroups and not by the conventional translation group this could provide insight into the macroscopic irreversibility paradox the induced semigroups are generated by fractional time derivatives of orders less than unity not by a first order time derivative as the conventional group invariance under the induced semiflows therefore leads to a new form of stationarity called fractional stationarity fractional stationarity provides the dynamical foundation for a generalized equilibrium concept

efficient algorithms for membership in boolean hierarchies of regular languages this paper provides efficient algorithms that decide membership for classes of several boolean hierarchies for which efficiency or even decidability were previously not known we develop new forbidden chain characterizations for the single levels of these hierarchies and obtain the following results the classes of the boolean hierarchy over level 1 of the dot depth hierarchy are decidable in nl previously only the decidability was known the same remains true if predicates mod d for fixed d are allowed if modular predicates for arbitrary d are allowed then the classes of the boolean hierarchy over level 1 are decidable for the restricted case of a two letter alphabet the classes of the boolean hierarchy over level 2 of the straubing therien hierarchy are decidable in nl this is the first decidability result for this hierarchy the membership problems for all mentioned boolean hierarchy classes are logspace many one hard for nl the membership problems for quasi aperiodic languages and for d quasi aperiodic languages are logspace many one complete for pspace

block operators and spectral discretizations every student of numerical linear algebra is familiar with block matrices and vectors the same ideas can be applied to the continuous analogues of operators functions and functionals it is shown here how the explicit consideration of block structures at the continuous level can be a useful tool in particular block operator diagrams lead to templates for spectral discretization of differential and integral equation boundary value problems in one space dimension by the rectangular differentiation identity and integration matrices introduced recently by driscoll and hale the templates are so simple that we are able to present them as executable matlab codes just a few lines long developing ideas through a sequence of 12 increasingly advanced examples the notion of the rectangular shape of a linear operator is made mathematically precise by the theory of fredholm operators and their indices and the block operator formulations apply to nonlinear problems too we propose the convention of representi

on nonlinear conservation laws with a nonlocal diffusion term abstract scalar one dimensional conservation laws with a nonlocal diffusion term corresponding to a riesz feller differential operator are considered solvability results for the cauchy problem in l are adapted from the case of a fractional derivative with homogeneous symbol the main interest of this work is the investigation of smooth shock profiles in the case of a genuinely nonlinear smooth flux function we prove the existence of such travelling waves which are monotone and satisfy the standard entropy condition moreover the dynamic nonlinear stability of the travelling waves under small perturbations is proven similarly to the case of the standard diffusive regularisation by constructing a lyapunov functional

fox h functions in fractional diffusion the h functions introduced by fox in 1961 are special functions of a very general nature which allow one to treat several phenomena including anomalous diffusion in a unified and elegant framework in this paper we express the fundamental solutions of the cauchy problem for the space time fractional diffusion equation in terms of proper fox h functions based on their mellin barnes integral representations we pay attention to the particular cases of space fractional time fractional and neutral fractional diffusion

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

generalized langevin equation with a three parameter mittag leffler noise abstract the relaxation functions for a given generalized langevin equation in the presence of a three parameter mittag leffler noise are studied analytically the results are represented by three parameter mittag leffler functions exact results for the velocity and displacement correlation functions of a diffusing particle are obtained by using the laplace transform method the asymptotic behavior of the particle in the short and long time limits are found by using the tauberian theorems it is shown that for large times the particle motion is subdiffusive for 1 and superdiffusive for many previously obtained results are recovered due to the many parameters contained in the noise term the model considered in this work may be used to improve the description of data and to model anomalous diffusive processes in complex media

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

first passage and arrival time densities for levy flights and the failure of the method of images we discuss the first passage time problem in the semi infinite interval for homogeneous stochastic markov processes with levy stable jump length distributions x l x 1 x l namely levy flights lfs in particular we demonstrate that the method of images leads to a result which violates a theorem due to sparre andersen according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density fptd governed by the universal long time decay t 3 2 conversely we show that for lfs the direct definition known from gaussian processes in fact defines the probability density of first arrival which for lfs differs from the fptd our findings are corroborated by numerical results

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

global attractors and steady state solutions for a class of reaction diffusion systems abstract we show that weak l p dissipativity implies strong l dissipativity and therefore implies the existence of global attractors for a general class of reaction diffusion systems this generalizes the results of alikakos and rothe the results on positive steady states especially for systems of three equations in our earlier work j differential equations 130 1996 59 91 are improved

the general time fractional wave equation for a vibrating string the solution of a general time fractional wave equation for a vibrating string is obtained in terms of the mittag leffler type functions and complete set of eigenfunctions of the sturm liouville problem the time fractional derivative used is taken in the caputo sense and the method of separation of variables and the laplace transform method are used to solve the equation some results for special cases of the initial and boundary conditions are obtained and it is shown that the corresponding solutions of the integer order equations are special cases of those of time fractional equations the proposed general equation may be used for modeling different processes in complex or viscoelastic media disordered materials etc

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

correlation functions for the fractional generalized langevin equation in the presence of internal and external noise we study generalized fractional langevin equations in the presence of a harmonic potential general expressions for the mean velocity and particle displacement the mean squared displacement position and velocity correlation functions as well as normalized displacement correlation function are derived we report exact results for the cases of internal and external friction that is when the driving noise is either internal and thus the fluctuation dissipation relation is fulfilled or when the noise is external the asymptotic behavior of the generalized stochastic oscillator is investigated and the case of high viscous damping overdamped limit is considered additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed in addition the cases of a constant external force and the force free case are obtained the validity of the generalized einstein relation for this process is discussed the considered fractional generalized langevin equation may be used to model anomalous diffusive processes including single file type diffusion

anomalous diffusion exact solution of the generalized langevin equation for harmonically bounded particle we study the effect of a disordered or fractal environment in the irreversible dynamics of a harmonic oscillator starting from a generalized langevin equation and using laplace analysis we derive exact expressions for the mean values variances and velocity autocorrelation function of the particle in terms of generalized mittag leffler functions the long time behaviors of these quantities are obtained and the presence of a whip back effect is analyzed

anomalous diffusion and the first passage time problem we study the distribution of the first passage time fpt in levy type anomalous diffusion using the recently formulated fractional fokker planck equation we obtain three results 1 we derive an explicit expression for the fpt distribution in terms of fox or h functions when the diffusion has zero drift 2 for the nonzero drift case we obtain an analytical expression for the laplace transform of the fpt distribution 3 we express the fpt distribution in terms of a power series for the case of two absorbing barriers the known results for ordinary diffusion brownian motion are obtained as special cases of our more general results

on a time space fractional backward diffusion problem with inexact orders abstract in this paper we focus on the backward diffusion problem with the caputo fractional derivative operator in time and a general spatial nonlocal operator for t 0 and s 0 t we consider the problem p s of recovering the distribution u x s from a measure of the final data u x t for the following non homogeneous time space fractional diffusion equation d t u x t k l u x t f x t in r n 0 t subject to the final condition u x t u t x in r n the derivative orders and the nonlocal operator are perturbed with noises firstly for 0 s t we prove the well posedness of problem p s by studying the unique existence and continuity with respect to the derivative orders the source term as well as the final value of the solution secondly for s 0 we verify the ill posedness of problem p 0 and use the method of modified iterated lavrentiev to construct a regularization solution from inexact data and inexact derivative orders we apply a modified form of the discrepancy principle to choose regularization parameter and establish new optimal convergence estimates between the exact solution and its regularized approximation

hilfer fractional advection diffusion equations with power law initial condition a numerical study using variational iteration method abstract we propose a hilfer advection diffusion equation of order 0 1 and type 0 1 and find the power series solution by using variational iteration method power series solutions are expressed in a form that is easy to implement numerically and in some particular cases solutions are expressed in terms of mittag leffler function absolute convergence of power series solutions is proved and the sensitivity of the solutions is discussed with respect to changes in the values of different parameters for power law initial conditions it is shown that the hilfer advection diffusion pde gives the same solutions as the caputo and riemann liouville advection diffusion pde to leading order the fractional solution compared to the non fractional solution increases rapidly with for 0 7 at a given time t but for 0 7 this factor is weakly sensitive to we also show that the truncation errors arising when using the partial sum as approximate solutions decay exponentially fast with the number of terms n used we find that for 0 7 the number of terms needed is weakly sensitive to the accuracy level and to the fractional order n 20 but for 0 7 the required number of terms increases rapidly with the accuracy level and also with the fractional order

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

fractional calculus with an integral operator containing a generalized mittag leffler function in the kernel in this paper we introduce and investigate a fractional calculus with an integral operator which contains the following family of generalized mittag leffler functions e a b c k z email protected n 0 c k email protected email protected z nn z b email protected c r a max 0 r k 1 r k 0 in its kernel l n being the familiar pochhammer symbol a number of corollaries and consequences of the main results presented here are also considered

convergence of the grunwald letnikov scheme for time fractional diffusion using bivariate generating functions we prove convergence of the grunwald letnikov difference scheme for the fractional diffusion equation in one space dimension with and without central linear drift in the fourier laplace domain as the space and time steps tend to zero in a well scaled way this implies convergence in distribution weak convergence of the discrete solution towards the probability of sojourn of a diffusing particle the difference schemes allow also interpretation as discrete random walks for fractional diffusion with central linear drift we show that in the fourier laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation

descriptional and computational complexity of finite automata a survey finite automata are probably best known for being equivalent to right linear context free grammars and thus for capturing the lowest level of the chomsky hierarchy the family of regular languages over the last half century a vast literature documenting the importance of deterministic nondeterministic and alternating finite automata as an enormously valuable concept has been developed in the present paper we tour a fragment of this literature mostly we discuss developments relevant to finite automata related problems like for example i simulation of and by several types of finite automata ii standard automata problems such as fixed and general membership emptiness universality equivalence and related problems and iii minimization and approximation we thus come across descriptional and computational complexity issues of finite automata we do not prove these results but we merely draw attention to the big picture and some of the main ideas involved

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

relaxation and diffusion models with non singular kernels abstract anomalous relaxation and diffusion processes have been widely quantified by fractional derivative models where the definition of the fractional order derivative remains a historical debate due to its limitation in describing different kinds of non exponential decays e g stretched exponential decay meanwhile many efforts by mathematicians and engineers have been made to overcome the singularity of power function kernel in its definition this study first explores physical properties of relaxation and diffusion models where the temporal derivative was defined recently using an exponential kernel analytical analysis shows that the caputo type derivative model with an exponential kernel cannot characterize non exponential dynamics well documented in anomalous relaxation and diffusion a legitimate extension of the previous derivative is then proposed by replacing the exponential kernel with a stretched exponential kernel numerical tests show that the caputo type derivative model with the stretched exponential kernel can describe a much wider range of anomalous diffusion than the exponential kernel implying the potential applicability of the new derivative in quantifying real world anomalous relaxation and diffusion processes

complexity of some problems from the theory of automata 1 1 given a finite alphabet l the regular events over x are those accepted by a finite state automaton by kleene s theorem a subset w of s is a regular event if and only if it can be constructed from the finiteword sets by boolean operations together with concatenation and operation in some sense the regular events are quite simple because they are accepted by a machine with no storage capacity nevertheless in recent years much attention has been paid to special subclasses of the class of regular events in this paper we shall be concerned with star free events events of dot depth one and piecewise testable events star free events are constructed like regular events from the finite word sets but with the restriction that the operation is not allowed events of dot depth one and piecewise testable events are star free of a very simple form and will be defined below star free events have been characterized in the work of schiitzenberger 1965 in terms of their syntactic monoid algebraic characterizations of the other two classes have been given simon 1975 knast 1983 from an algorithmic point of view these algebraic characterizations do not yield efficient procedures because computing the syntactic monoid of a regular event given e g by an automaton is obviously time consuming in this paper we investigate the complexity of three problems which we now describe in the style of garey and johnson 1979 we refer the reader to aho hopcroft and ullman 1983 garey and johnson 1979 for standard concepts of complexity theory

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

generalized cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator abstract this paper is devoted to proving the existence and uniqueness of solutions to cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions an approach based on the equivalence of the nonlinear cauchy type problem to a nonlinear volterra integral equation of the second kind and applying a variant of the banach s fixed point theorem to prove uniqueness and existence of the solution is presented the cauchy type problems for integro differential equations of volterra type with composite fractional derivative operator which contain the generalized mittag leffler function in the kernel are considered using the method of successive approximation and the laplace transform method explicit solutions of the open problem proposed by srivastava and tomovski 2009 11 are established in terms of the multinomial mittag leffler function

levy processes on a generalized fractal comb comb geometry constituted of a backbone and fingers is one of the most simple paradigm of a two dimensional structure where anomalous diffusion can be realized in the framework of markov processes however the intrinsic properties of the structure can destroy this markovian transport these effects can be described by the memory and spatial kernels in particular the fractal structure of the fingers which is controlled by the spatial kernel in both the real and the fourier spaces leads to the levy processes levy flights and superdiffusion this generalization of the fractional diffusion is described by the riesz space fractional derivative in the framework of this generalized fractal comb model levy processes are considered and exact solutions for the probability distribution functions are obtained in terms of the fox h function for a variety of the memory kernels and the rate of the superdiffusive spreading is studied by calculating the fractional moments for a special form of the memory kernels we also observed a competition between long rests and long jumps finally we considered the fractal structure of the fingers controlled by a weierstrass function which leads to the power law kernel in the fourier space this is a special case when the second moment exists for superdiffusion in this competition between long rests and long jumps

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

effects of a fractional friction with power law memory kernel on string vibrations in this paper we give an analytical treatment of a wave equation for a vibrating string in the presence of a fractional friction with power law memory kernel the exact solution is obtained in terms of the mittag leffler type functions and a generalized integral operator containing a four parameter mittag leffler function in the kernel the method of separation of the variables laplace transform method and sturm liouville problem are used to solve the equation analytically the asymptotic behaviors of the solution of a special case fractional differential equation are obtained directly from the analytical solution of the equation and by using the tauberian theorems the proposed model may be used for describing processes where the memory effects of complex media could not be neglected

the wright functions as solutions of the time fractional diffusion equation we revisit the cauchy problem for the time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order email protected 0 2 by using the fourier laplace transforms the fundamentals solutions green functions are shown to be high transcendental functions of the wright type that can be interpreted as spatial probability density functions evolving in time with similarity properties we provide a general representation of these functions in terms of mellin barnes integrals useful for numerical computation

from cascade decompositions to bit vector algorithms a vector algorithm is an algorithm that applies a bounded number of vector operations to an input vcctor regardless of the length of the input in this paper we describe the links between the existence of vector algorithms and the cascade decompositions of counter free automata we show that any computation that can be carried out with a counter free automaton can be recast as a vector algorithm moreover we show that for a class of automata that is closely related to algorithms in bio computing the complexity of the resulting algorithms is linear in the number of transitions of the original automaton

diffusion on random site percolation clusters theory and nmr microscopy experiments with model objects quasi two dimensional random site percolation model objects were fabricated based on computer generated templates samples consisting of two compartments a reservoir of mathrm h 2 mathrm o gel attached to a percolation model object which was initially filled with mathrm d 2 mathrm o were examined with nuclear magnetic resonance microscopy for rendering proton spin density maps the propagating proton deuteron interdiffusion profiles were recorded and evaluated with respect to anomalous diffusion parameters the deviation of the concentration profiles from those expected for unobstructed diffusion directly reflects the anomaly of the propagator for diffusion on a percolation cluster the fractal dimension of the random walk d w evaluated from the diffusion measurements on the one hand and the fractal dimension d f deduced from the spin density map of the percolation object on the other permits one to experimentally compare dynamical and static exponents approximate calculations of the propagator are given on the basis of the fractional diffusion equation furthermore the ordinary diffusion equation was solved numerically for the corresponding initial and boundary conditions for comparison the anomalous diffusion constant was evaluated and is compared to the brownian case some ad hoc correction of the propagator is shown to pay tribute to the finiteness of the system in this way anomalous solutions of the fractional diffusion equation could experimentally be verified

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

some complexity results for polynomial rational expressions it is known that any languages of level 3 2 of the straubing therien hierarchy can be written as a polynomial expression that is a finite union of languages of the form a0 b1a 1 b kak where the ai and bi are subsets of the alphabet in this paper we prove that such a polynomial expression for a 3 2 level language can be of exponential size more precisely we exhibit an n state minimal automaton recognizing a level 3 2 language for which the shortest polynomial expresion has exponential size in n

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

a polynomial time algorithm for the local testability problem of deterministic finite automata the local testability problem of deterministic finite automata is investigated a locally testable language is a language with the property that for some nonnegative integer k whether or not a word w is in the language depends on 1 the prefix and suffix of w of length k and 2 the set of substrings of w length k 1 without regard to the order in which these substrings occur the local testability problem is given a deterministic finite automation to decide whether or not it accepts a locally testable language the authors present an o n sup 2 time algorithm for the local testability problem based on two simple properties that characterize locally testable automata

generalized diffusion equation with fractional derivatives within renyi statistics by using the zubarev nonequilibrium statistical operator method and the liouville equation with fractional derivatives a generalized diffusion equation with fractional derivatives is obtained within the renyi statistics averaging in generalized diffusion coefficient is performed with a power distribution with the renyi parameter q

characterizations of some classes of regular events abstract we study two special classes of regular events the piecewise testable events and the events of dot depth one several characterizations of these events are given especially in terms of their minimal automaton

ergodic transitions in continuous time random walks we consider continuous time random walk models described by arbitrary sojourn time probability density functions we find a general expression for the distribution of time averaged observables for such systems generalizing some recent results presented in the literature for the case where sojourn times are identically distributed independent random variables our results shed some light on the recently proposed transitions between ergodic and weakly nonergodic regimes on the other hand for the case of nonidentical trapping time densities over the lattice points the distribution of time averaged observables reveals that such systems are typically nonergodic in agreement with some recent experimental evidences on the statistics of blinking quantum dots some explicit examples are considered in detail our results are independent of the lattice topology and dimensionality

modeling diffusive transport with a fractional derivative without singular kernel abstract in this paper we present an alternative representation of the diffusion equation and the diffusion advection equation using the fractional calculus approach the spatial time derivatives are approximated using the fractional definition recently introduced by caputo and fabrizio in the range 0 2 for the space and time domain respectively in this representation two auxiliary parameters x and t are introduced these parameters related to equation results in a fractal space time geometry provide an entire new family of solutions for the diffusion processes the numerical results showed different behaviors when compared with classical model solutions in the range 0 1 the concentration exhibits the non markovian levy flights and the subdiffusion phenomena when 1 the classical case is recovered when 1 2 the concentration exhibits the markovian levy flights and the superdiffusion phenomena finally when 2 the concentration is anomalous dispersive and we found ballistic diffusion

stochastic calculus for uncoupled continuous time random walks the continuous time random walk ctrw is a pure jump stochastic process with several applications not only in physics but also in insurance finance and economics a definition is given for a class of stochastic integrals driven by a ctrw which includes the it and stratonovich cases an uncoupled ctrw with zero mean jumps is a martingale it is proved that as a consequence of the martingale transform theorem if the ctrw is a martingale the it integral is a martingale too it is shown how the definition of the stochastic integrals can be used to easily compute them by monte carlo simulation the relations between a ctrw its quadratic variation its stratonovich integral and its it integral are highlighted by numerical calculations when the jumps in space of the ctrw have a symmetric levy alpha stable distribution and its waiting times have a one parameter mittag leffler distribution remarkably these distributions have fat tails and an unbounded quadratic variation in the diffusive limit of vanishing scale parameters the probability density of this kind of ctrw satisfies the space time fractional diffusion equation fde or more in general the fractional fokker planck equation which generalizes the standard diffusion equation solved by the probability density of the wiener process and thus provides a phenomenologic model of anomalous diffusion we also provide an analytic expression for the quadratic variation of the stochastic process described by the fde and check it by monte carlo

generalized distributed order diffusion equations with composite time fractional derivative abstract in this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the fourier laplace transform method we represent solutions in terms of infinite series in fox h functions the fractional and second moments are derived by using mittag leffler functions we observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives generalized uniformly distributed order diffusion equation as a model for strong anomalous behavior is analyzed by using tauberian theorem some previously obtained results are special cases of those presented in this paper

pspace complete problems for subgroups of free groups and inverse finite automata we investigate the complexity of algorithmic problems on finitely generated subgroups of free groups margolis and meakin showed how a finite monoid synt h can be canonically and effectively associated with such a subgroup h we show that h is pure that is closed under radical if and only if synt h is aperiodic we also show that testing for this property of h is pspace complete in the process we show that certain problems about finite automata which are pspace complete in general remain pspace complete when restricted to injective and inverse automata with single accept state whereas they are known to be in nc for permutation automata with single accept state

fractional diffusion equation with a generalized riemann liouville time fractional derivative in this paper the solution of a fractional diffusion equation with a hilfer generalized riemann liouville time fractional derivative is obtained in terms of mittag leffler type functions and fox s h function the considered equation represents a quite general extension of the classical diffusion heat conduction equation the methods of separation of variables laplace transform and analysis of the sturm liouville problem are used to solve the fractional diffusion equation defined in a bounded domain by using the fourier laplace transform method it is shown that the fundamental solution of the fractional diffusion equation with a generalized riemann liouville time fractional derivative defined in the infinite domain can be expressed via fox s h function it is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the caputo and riemann liouville sense are special cases of those diffusion equations with the hilfer generalized riemann liouville time fractional derivative the asymptotic behaviour of the solutions are found for large values of the spatial variable the fractional moments of the fundamental solution of the fractional diffusion equation are obtained the obtained results are relevant in the context of glass relaxation and aquifer problems

average time spent by levy flights and walks on an interval with absorbing boundaries we consider a levy flyer of order alpha that starts from a point x 0 on an interval o l with absorbing boundaries we find a closed form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed these two quantities are equivalent to the mean first passage times for levy flights and levy walks respectively using fractional differential equations with a riesz kernel we find exact analytical expressions for both quantities in the continuous limit we show that numerical solutions for the discrete levy processes converge to the continuous approximations in all cases except the case of alpha 2 and the cases of x 0 0 and x 0 l for alpha 2 when the second moment of the flight length distribution exists our result is replaced by known results of classical diffusion we show that if x 0 is placed in the vicinity of absorbing boundaries the average total length has a minimum at alpha 1 corresponding to the cauchy distribution we discuss the relevance of this result to the problem of foraging which has received recent attention in the statistical physics literature

some results on the generalized star height problem abstract we prove some results related to the generalized star height problem in this problem as opposed to the restricted star height problem complementation is considered as a basic operator we first show that the class of languages of star height n is closed under certain operations left and right quotients inverse alphabetic morphisms injective star free substitutions it is known that languages recognized by a commutative group are of star height 1 we extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by z 2 z n in the same direction we show that one of the languages that were conjectured to be of star height 2 during the past ten years is in fact of star height 1 next we show that if a rational language l is recognized by a monoid of the variety generated by wreath products of the form m g n where m and n are aperiodic monoids and g is a commutative group the l is of star height 1 finally we show that every rational language is the inverse image under some morphism between free monoids of a language of resticted star height 1

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

leapover lengths and first passage time statistics for levy flights exact results for the first passage time and leapover statistics of symmetric and one sided levy flights lfs are derived lfs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one sided lfs and surprisingly with an index alpha 2 for symmetric lfs the first passage time distribution scales like a power law with an index 1 2 as required by the sparre andersen theorem for symmetric lfs whereas one sided lfs have a narrow distribution of first passage times the exact analytic results are confirmed by extensive simulations

on fractional partial differential equations related to quantum mechanics in this paper we investigate the solutions of generalized fractional partial differential equations involving the caputo time fractional derivative and the liouville space fractional derivatives the solutions of these equations are obtained by employing the joint laplace and fourier transforms several special cases as solutions of one dimensional non homogeneous fractional equations occurring in quantum mechanics are presented in the concluding section the results given earlier by debnath 2003 fract calc appl anal 6 119 55 saxena et al 2010 appl math comput 216 1412 7 and pagnini and mainardi 2010 j comput appl math 233 1590 5 follow as special cases of our findings