towards a physics on fractals differential vector calculus in three dimensional continuum with fractal metric abstract one way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric on this way different definitions of the fractal metric were suggested to account for the essential fractal features in this work we develop the metric differential vector calculus in a three dimensional continuum with a non euclidean metric the metric differential forms and laplacian are introduced fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the moisil teodoresco operator which has been introduced and partially developed in this paper the relations between the metric and conventional operators are revealed it should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric this offers a novel tool to study physics on fractals

equations of motion in a non integer dimensional space equations of motion are derived for a fractional dimensional system of n spatial coordinates to be used as an effective description of anisotropic and confined systems an existing measure theoretic approach is extended to multiple variables and different degrees of confinement in orthogonal directions and comparisons are made with the analytic continuation of gaussian integrals this is applied to the variational principle and equations of motion for a field described by a lagrange density are found a specific example is looked at in schrodinger wave mechanics particularly in three coordinate systems

the randomization technique as a modeling tool and solution procedure for transient markov processes we present a randomization procedure for computing transient solutions to discrete state space continuous time markov processes this procedure computes transient state probabilities it is based on a construction relating a continuous time markov process to a discrete time markov chain modifications and extensions of the randomization method allow for computation of distributions of first passage times and sojourn times in markov processes and also the computation of expected cumulative occupancy times and expected number of events occurring during a time interval several implementations of the randomization procedure are discussed in particular we present an implementation for a general class of markov processes that can be described in terms of state space s event set e rate vectors r and target vectors t abbreviated as sert this general approach can handle systems whose state spaces are quite large if they have sparse generators

fractional integral and differential equations for a class of levy type probability densities an application of the fractional calculus to a class of levy distribution functions leads to the conclusion that the levy index fractal dimension mu is identical to the order of the fractional liouville riemann integral operator the corresponding fractional integral and differential equations are presented and solutions of levy type one sided probability densities are given and discussed

anisotropic fractal media by vector calculus in non integer dimensional space a review of different approaches to describe anisotropic fractal media is proposed in this paper differentiation and integration non integer dimensional and multi fractional spaces are considered as tools to describe anisotropic fractal materials and media we suggest a generalization of vector calculus for non integer dimensional space by using a product measure method the product of fractional and non integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models the integration over non integer dimensional spaces is considered in this paper differential operators of first and second orders for fractional space and non integer dimensional space are suggested the differential operators are defined as inverse operations to integration in spaces with non integer dimensions non integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media the poisson s equation for fractal medium the euler bernoulli fractal beam and the timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media

fractal geometry in solids and structures abstract the present paper presents certain methods for the consideration of the influence of the fractal geometry in solid and structural mechanics the method of fractal interpolation function is introduced and the fractals are considered as the fixed points of a given set valued transformation an attempt is made here to define the mechanical quantities on fractal sets using some results of the theory of besov spaces then we try to extend the classical fem for the case of fractal bodies and fractal boundaries then corresponding error estimates are derived the methods of fractal analysis permit the treatment of complicated or yet unsolved problems in engineering

recursive markov chains stochastic grammars and monotone systems of nonlinear equations we define recursive markov chains rmcs a class of finitely presented denumerable markov chains and we study algorithms for their analysis informally an rmc consists of a collection of finite state markov chains with the ability to invoke each other in a potentially recursive manner rmcs offer a natural abstract model for probabilistic programs with procedures they generalize in a precise sense a number of well studied stochastic models including stochastic context free grammars scfg and multi type branching processes mt bp we focus on algorithms for reachability and termination analysis for rmcs what is the probability that an rmc started from a given state reaches another target state or that it terminatesq these probabilities are in general irrational and they arise as least fixed point solutions to certain monotone systems of nonlinear equations associated with rmcs we address both the qualitative problem of determining whether the probabilities are 0 1 or in between and the quantitative problems of comparing the probabilities with a given bound or approximating them to desired precision we show that all these problems can be solved in pspace using a decision procedure for the existential theory of reals we provide a more practical algorithm based on a decomposed version of multi variate newton s method and prove that it always converges monotonically to the desired probabilities we show this method applies more generally to any monotone polynomial system we obtain polynomial time algorithms for various special subclasses of rmcs among these for scfgs and mt bps equivalently for 1 exit rmcs the qualitative problem can be solved in p time for linearly recursive rmcs the probabilities are rational and can be computed exactly in p time we show that our pspace upper bounds cannot be substantially improved without a breakthrough on long standing open problems the square root sum problem and an arithmetic circuit decision problem that captures p time on the unit cost rational arithmetic ram model we show that these problems reduce to the qualitative problem and to the approximation problem to within any nontrivial error for termination probabilities of general rmcs and to the quantitative decision problem for termination extinction of scfgs mt bps

a simple empirical formula of origin intensity factor in singular boundary method for two dimensional hausdorff derivative laplace equations with dirichlet boundary abstract this paper presents a simple empirical formula of origin intensity factor in singular boundary method sbm solution of hausdorff derivative laplace equations the sbm with the empirical formula is mathematically more simple and computationally more efficient than using the other techniques for origin intensity factor numerical experiments simulate the steady heat conduction through fractal media governed by the hausdorff laplace equation and show the efficiency and reliability benefits of the present sbm empirical formula

local fractional fokker planck equation we propose a new class of differential equations which we call local fractional differential equations they involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time a local fractional analog of the fokker planck equation has been derived starting from the chapman kolmogorov condition we solve the equation with a specific choice of the transition probability and show how subdiffusive behavior can arise

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

derivation and solutions of some fractional black scholes equations in coarse grained space and time application to merton s optimal portfolio abstract by using the new fractional taylor s series of fractional order f x h e h d x f x where e denotes the mittag leffler function and d x is the so called modified riemann liouville fractional derivative which we introduced recently to remove the effects of the non zero initial value of the function under consideration one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a usual gaussian white noise one can then derive two new families of fractional black scholes equations and one shows how one can obtain their solutions merton s optimal portfolio is once more considered and some new results are contributed with respect to the modeling on one hand and to the solution on the other hand finally one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics

smooth stable invariant manifolds and arbitrary growth rates abstract for nonautonomous linear equations v a t v with a generalized exponential dichotomy we show that there is a smooth stable invariant manifold for the perturbed equation v a t v f t v provided that f is sufficiently small the generalized exponential dichotomies may exhibit stable and unstable behaviors with respect to arbitrary growth rates e c t for some function t we consider the general case of nonuniform exponential dichotomies and the result is obtained in banach spaces moreover we show that for an equivariant system the dynamics on the stable manifold in a certain class of graphs is also equivariant we emphasize that this result cannot be obtained by averaging over the symmetry

csl model checking algorithms for qbds we present an in depth treatment of model checking algorithms for a class of infinite state continuous time markov chains known as quasi birth death processes the model class is described in detail as well as the logic csl to express properties of interest using a new property independency concept we provide model checking algorithms for all the csl operators special emphasis is given to the time bounded until operator for which we present a new and efficient computational procedure named uniformization with representatives by the use of an application driven dynamic stopping criterion the algorithm stops whenever the property to be checked can be certified or falsified a comprehensive case study of a connection management system shows the versatility of our new algorithms

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

fractal solids product measures and fractional wave equations this paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass or spatial fractal dimension d a surface fractal dimension d and a resolution length scale r the focus is on pre fractal media i e those with lower and upper cut offs through a theory based on a dimensional regularization in which d is also the order of fractional integrals employed to state global balance laws in effect the governing equations are cast in forms involving conventional integer order integrals while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving d d and r this procedure allows a specification of a geometry configuration of continua by fractal metric coefficients on which the continuum mechanics is subsequently constructed while all the derived relations depend explicitly on d d and r upon setting d 3 and d 2 they reduce to conventional forms of governing equations for continuous media with euclidean geometries whereas the original formulation was based on a riesz measure and thus more suited to isotropic media the new model is based on a product measure making it capable of grasping local fractal anisotropy finally the one two and three dimensional wave equations are developed showing that the continuum mechanics approach is consistent with that obtained via variational energy principles

a scaling method and its applications to problems in fractional dimensional space a scaling method is proposed to find 1 the volume and the surface area of a generalized hypersphere in a fractional dimensional space and 2 the solid angle at a point for the same space it is demonstrated that the total dimension of the fractional space can be obtained by summing the dimension of the fractional line element along each axis the regularization condition is defined for functions depending on more than one variable this condition is applied 1 to find a closed form expression for the fractional gaussian integral 2 to establish a relationship between a fractional dimensional space and a fractional integral 3 to develop the bochner theorem and 4 to obtain an expression for the fractional integral of the mittag leffler function some possible extensions of this work are also discussed

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

functional characterization of generalized langevin equations we present an exact functional formalism to deal with linear langevin equations with arbitrary memory kernels and driven by an arbitrary noise structure characterized through its characteristic functional no other hypothesis is assumed over the noise neither do we use the fluctuation dissipation theorem we find that the characteristic functional of the linear process can be expressed in terms of noise functional and the green function of the deterministic memory like dissipative dynamics this yields a procedure for calculating the full kolmogorov hierarchy of the non markov process as examples we have characterized through the 1 time probability a noise induced interplay between the dissipative dynamics and the structure of different noises conditions that lead to non gaussian statistics and distributions with long tails are analysed the introduction of arbitrary fluctuations in fractional langevin equations has also been pointed out

common aspects of q deformed lie algebras and fractional calculus fractional calculus and q deformed lie algebras are closely related both concepts expand the scope of standard lie algebras to describe generalized symmetries a new class of fractional q deformed lie algebras is proposed which for the first time allows a smooth transition between different lie algebras for the fractional harmonic oscillator the corresponding fractional q number is derived it is shown that the resulting energy spectrum is an appropriate tool to describe e g the ground state spectra of even even nuclei in addition the equivalence of rotational and vibrational spectra for fractional q deformed lie algebras is shown and the b e2 values for the fractional q deformed symmetric rotor are calculated a first interpretation of half integer representations of the fractional rotation group is given in terms of a description of k 1 2 band spectra of odd even nuclei

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

quaternionic time harmonic maxwell operator the theory of quaternionic alpha hyperholomorphic functions synonyms monogenic regular spatial holomorphic vectors is currently developing rapidly in particular many integral formulae with explicit reproducing kernels have been obtained in this work we establish a one to one correspondence between time harmonic monochromatic electromagnetic fields and pairs of mutually conjugate hyperholomorphic functions this leads to the cauchy type integral associated with maxwell s equations some main integral formulae for maxwell s equations involving this cauchy type integral are obtained it should be mentioned that in fact we introduce and study a somewhat more general quaternionic object which has better algebraic and analytic properties than the physical maxwell operator and which contains the latter as a special case

exact discretization of fractional laplacian abstract an exact discretization of fractional order laplacian for n dimensional space is suggested particular solutions of fractional order partial difference equations with the proposed discrete laplace operators are suggested

tsallis entropy induced metrics and cat k spaces abstract generalizing the group structure of the euclidean space we construct a riemannian metric on the deformed set r q n induced by the tsallis entropy composition property we show that the tsallis entropy is a hyperbolic analogue of the euclidean boltzmann gibbs shannon entropy and find a geometric interpretation for the nonextensive parameter q we provide a geometric explanation of the uniqueness of the tsallis entropy as reflected through its composition property which is provided by the abe and the santos axioms for two or more interacting systems described by the tsallis entropy having different values of q we argue why a suitable extension of this construction is provided by the cartan alexandrov toponogov metric spaces with a uniform negative curvature upper bound

fractional dirac bracket and quantization for constrained systems so far it is not well known how to deal with dissipative systems there are many paths of investigation in the literature and none of them present a systematic and general procedure to tackle the problem on the other hand it is well known that the fractional formalism is a powerful alternative when treating dissipative problems in this paper we propose a detailed way of attacking the issue using fractional calculus to construct an extension of the dirac brackets in order to carry out the quantization of nonconservative theories through the standard canonical way we believe that by using the extended dirac bracket definition it will be possible to analyze more deeply gauge theories starting with second class systems

anomalous diffusion modeling by fractal and fractional derivatives this paper makes an attempt to develop a fractal derivative model of anomalous diffusion we also derive the fundamental solution of the fractal derivative equation for anomalous diffusion which characterizes a clear power law this new model is compared with the corresponding fractional derivative model in terms of computational efficiency diffusion velocity and heavy tail property the merits and distinctions of these two models of anomalous diffusion are then summarized

quantum field theory models on fractal spacetime i introduction and overview deux types de modeles de champs scalaires sont consideres l un a son propagateur donne par un temps propre ou une representation chaleur noyau l autre a un propagateur hierarchique la version du reseau fractal du propagateur a temps propre est une reflexion positive

a possible deformed algebra and calculus inspired in nonextensive thermostatistics we present a deformed algebra related to the q exponential and the q logarithm functions that emerge from nonextensive statistical mechanics we also develop a q derivative and consistently a q integral for which the q exponential is an eigenfunction the q derivative and the q integral have a dual nature that is also presented

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

variational approach and deformed derivatives recently we have demonstrated that there exists a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely the tsallis framework and the kaniadakis scenario with a local form of fractional derivative operators for fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure here in this paper we present an extension of the traditional calculus of variations for systems containing deformed derivatives embedded into the lagrangian and the lagrangian densities for classical and field systems the results extend the classical euler lagrange equations and the hamiltonian formalism the resulting dynamical equations seem to be compatible with those found in the literature specially with mass dependent and with nonlinear equations for systems in classical and quantum mechanics examples are presented to illustrate applications of the formulation also the conserved noether current is worked out

inhomogeneity and complexity measures for spatial patterns abstract in this work we examine two different measures for inhomogeneity and complexity that are derived from non extensive considerations a la tsallis their performance is then tested on theoretically generated patterns all measures are found to exhibit a most sensitive behaviour for sierpinski carpets the procedures here introduced provide us with new powerful tsallis tools for analysing the inhomogeneity and complexity of spatial patterns

discussing the extension and applications of a variational approach with deformed derivatives one has recently presented an extension of the standard variational calculus to include the presence of deformed derivatives both in the lagrangian of systems of particles and in the lagrangian density of field theoretic models classical euler lagrange equations and the hamiltonian formalism have been reassessed in this approach whenever applied to a number of physical systems the resulting dynamical equations come out to be the correct ones found in the literature especially with mass dependent and with nonlinear equations for classical and quantum mechanical systems in the present contribution one extends the variational approach including a piecewise form of deformed derivatives to study higher order dissipative systems and to obtain as an option deformed equations as well applications to concrete situations are contemplated such as an accelerated point charge this is the problem of the abraham lorentz dirac force stochastic dynamics like the langevin the advection convection reaction and fokker planck equations the korteweg de vries equation the landau lifshitz gilbert problem and the caldirola kanai hamiltonian and heat transfer equation of the fourier and non fourier types by considering these different examples it is shown that the formulation proposed in this paper may be a simple but promising path for dealing for example with dissipative nonlinear stochastic systems and the anomalous heat transfer problem by adopting a variational approach one has recently presented an extension of the standard variational calculus to include the presence of deformed derivatives both in the lagrangian of systems of particles and in the lagrangian density of field theoretic models classical euler lagrange equations and the hamiltonian formalism have been reassessed in this approach whenever applied to a number of physical systems the resulting dynamical equations come out to be the correct ones found in the literature especially with mass dependent and with nonlinear equations for classical and quantum mechanical systems in the present contribution one extends the variational approach including a piecewise form of deformed derivatives to study higher order dissipative systems and to obtain as an option deformed equations as well applications to concrete situations are contemplated such as an accelerated point charge this is the problem of the abraham lorentz dirac force stochastic dynamics like the langevin the advection convection reaction and

quantum field theory on fractal spacetime a new regularisation method spacetime is modelled as a fractal subset of rn analysis on homogeneous sets with non integer hausdorff dimensions is applied to the low order perturbative renormalisation of quantum electrodynamics this new regularisation method implements the dirac matrices and tensors in r4 without difficulties is gauge invariant covariant and differs from dimensional regularisation in some aspects

non linear kinetics underlying generalized statistics abstract the purpose of the present effort is threefold firstly it is shown that there exists a principle that we call kinetical interaction principle kip underlying the non linear kinetics in particle systems independently on the picture kramers boltzmann used to describe their time evolution secondly the kip imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy thirdly the kip allows on one hand to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and on the other hand suggests how we can introduce naturally new distributions finally as a working example of the approach to the non linear kinetics here presented a new non extensive statistics is constructed and studied starting from a one parameter deformation of the exponential function holding the relation f x f x 1

on calculus of local fractional derivatives abstract local fractional derivative lfd operators have been introduced in the recent literature chaos 6 1996 505 513 being local in nature these derivatives have proven useful in studying fractional differentiability properties of highly irregular and nowhere differentiable functions in the present paper we prove leibniz rule chain rule for lfd operators generalization of directional lfd and multivariable fractional taylor series to higher orders have been presented

fractional calculus as a macroscopic manifestation of randomness we generalize the method of van hove physica amsterdam 21 517 1955 so as to deal with the case of nonordinary statistical mechanics that being phenomena with no time scale separation we show that in the case of ordinary statistical mechanics even if the adoption of the van hove method imposes randomness upon hamiltonian dynamics the resulting statistical process is described using normal calculus techniques on the other hand in the case where there is no time scale separation this generalized version of van hove s method not only imposes randomness upon the microscopic dynamics but it also transmits randomness to the macroscopic level as a result the correct description of macroscopic dynamics has to be expressed in terms of the fractional calculus

williams watts dielectric relaxation a fractal time stochastic process dielectric relaxation in amorphous materials is treated in a defect diffusion model where relaxation occurs when a mobile defect such as a vacancy reaches a frozen in dipole the random motion of the defect is assumed to be governed by a fractal time stochastic process where the mean duration between defect movements is infinite when there are many more defects than dipoles the williams watts decaying fractional exponential relaxation law is derived the argument of the exponential is related to the number of distinct sites visited by the random walk of the defect for the same reaction dynamics but with more traps than walkers an algebraically decaying relaxation is found

on the local fractional derivative abstract we present the necessary conditions for the existence of the kolwankar gangal local fractional derivatives kg lfd and introduce more general but weaker notions of lfds by using limits of certain integral averages of the difference quotient by applying classical results due to stein and zygmund 1965 16 we show that the kg lfd is almost everywhere zero in any given intervals we generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

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

a new definition of fractional derivative we give a new definition of fractional derivative and fractional integral the form of the definition shows that it is the most natural definition and the most fruitful one the definition for email protected email protected 1 coincides with the classical definitions on polynomials up to a constant further if a 1 the definition coincides with the classical definition of first derivative we give some applications to fractional differential equations

on uniform exponential stability for skew evolution semiflows on banach spaces the aim of this paper is to give several characterizations for the property of stability for skew evolution semiflows on banach spaces thus we obtain generalizations of some well known results due to barbashin datko and rolewicz in the case of evolution equations in banach spaces a unified treatment in the uniform setting is provided