moduli spaces of lumps on real projective space harmonic maps that minimise the dirichlet energy in their homotopy classes are known as lumps lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement this has consequences for the behaviour of lumps and their symmetries an interesting feature is that the moduli space of charge three lumps is a 7 dimensional manifold of cohomogeneity one which can be described as a one parameter family of symmetry orbits of d2 symmetric maps in this paper we discuss the charge three moduli spaces of lumps from two perspectives discrete symmetries of lumps and the riemann hurwitz formula we then calculate the metric and find explicit formulas for various geometric quantities we also discuss the implications for lump decay

fuzzy geometry an updated overview abstract this paper briefly reviews work on the fuzzy topology and geometry of image subsets including adjacency separation and connectedness distance and relative position area perimeter and diameter convexity medial axes and thinning as well as some applications of these concepts in image processing and analysis

particle tracking for fractional diffusion with two time scales abstract previous work y zhang m m meerschaert b baeumer particle tracking for time fractional diffusion phys rev e 78 2008 036705 showed how to solve time fractional diffusion equations by particle tracking this paper extends the method to the case where the order of the fractional time derivative is greater than one a subordination approach treats the fractional time derivative as a random time change of the corresponding cauchy problem with a first derivative in time one novel feature of the time fractional case of order greater than one is the appearance of clustering in the operational time subordinator which is non markovian solutions to the time fractional equation are probability densities of the underlying stochastic process the process models movement of individual particles the evolution of an individual particle in both space and time is captured in a pair of stochastic differential equations or langevin equations monte carlo simulation yields particle location and the ensemble density approximates the solution to the variable coefficient time fractional diffusion equation in one or several spatial dimensions the particle tracking code is validated against inverse transform solutions in the simplest cases further applications solve model equations for fracture flow and upscaling flow in complex heterogeneous porous media these variable coefficient time fractional partial differential equations in several dimensions are not amenable to solution by any alternative method so that the grid free particle tracking approach presented here is uniquely appropriate

spectral analysis of fractional kinetic equations with random data we present a spectral representation of the mean square solution of the fractional kinetic equation also known as fractional diffusion equation with random initial condition gaussian and non gaussian limiting distributions of the renormalized solution of the fractional in time and in space kinetic equation are described in terms of multiple stochastic integral representations

first passage times of levy flights coexisting with subdiffusion we investigate both analytically and numerically the first passage time fpt problem in one dimension for anomalous diffusion processes in which l evy flights and subdiffusion coexist we analyze the fpt for three subclasses of l evy stable motions i symmetric l evy motions characterized by l evy index ensuremath mu 0l ensuremath mu l2 and skewness parameter ensuremath beta 0 ii one sided l evy motions with ensuremath mu 0l ensuremath mu l1 and skewness ensuremath beta 1 and iii two sided skewed l evy motions the extreme case 1l ensuremath mu l2 and skewness ensuremath beta ensuremath 1 in all three cases the waiting times between successive jumps are heavy tailed with index ensuremath alpha we show that in all three cases the fpt distributions are power laws our findings extend earlier studies on fpts of l evy flights by considering the interplay between long rests and the l evy long jumps

ulam hyers stability of fractional langevin equations in this paper we discuss ulam hyers stability of nonlinear fractional langevin equations by using the boundedness monotonicity and nonnegative properties of classical and generalized mittag leffler functions further the same problem for the associated impulsive systems is investigated finally an example is given to illustrate our theoretical results

arithmetic brownian motion subordinated by tempered stable and inverse tempered stable processes abstract in the last decade the subordinated processes have become popular and have found many practical applications therefore in this paper we examine two processes related to time changed subordinated classical brownian motion with drift called arithmetic brownian motion the first one so called normal tempered stable is related to the tempered stable subordinator while the second one to the inverse tempered stable process we compare the main properties such as probability density functions laplace transforms ensemble averaged mean squared displacements of such two subordinated processes and propose the parameters estimation procedures moreover we calibrate the analyzed systems to real data related to indoor air quality

uncoupled continuous time random walk finite jump length probability density function in this work the uncoupled continuous time random walk model with the exponential jump length probability density function pdf is considered exact solutions for the pdf and second moment are obtained and investigated for different kinds of waiting time pdf

coupled continuous time random walk approach to the rachev ruschendorf model for financial data abstract in this paper we expand the rachev ruschendorf asset pricing model introducing a coupled continuous time random walk ctrw like form of the random number of price changes such a form results from the concept of the random clustering procedure that resembles the coarse graining methods of statistical physics and on the other hand indicates applicability of the ctrw idea widely used in physics to model anomalous diffusion for describing financial markets in the framework of the proposed model we derive the limiting distributions of log returns and the corresponding pricing formulas for european call option in order to illustrate the obtained theoretical results we present their fitting with several sets of financial data

fokker planck type equations associated with subordinated processes controlled by tempered α stable processes in this paper we introduce two models of stochastic processes driven by brownian motion and fractional brownian motion subordinated with tempered stable waiting times by using a new integro differential operator we obtain the generalized fokker planck type equations associated with these subordinated stochastic processes

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

on machine recognition of hand painted chinese characters by feature relaxation abstract a new relaxation matching method based on features is introduced for the recognition of hand printed chinese characters the types of features are selected carefully to reflect the structural information of characters matching probabilities between two features one from the mask and the other from input are computed by the relaxation method a new distance measure between two characters based on these matching probabilities is defined we demonstrate through examples the utility of the new approach in the recognition of hand printed chinese characters it is especially powerful in distinguishing similarly shaped characters within a cluster produced by preclassification

a mathematical theory of planar particle channeling in crystals abstract we use a simple two phase averaging method for odes to characterize the class of energetic positively charged particle motions in the relativistic perfect crystal model that remain uniformly close to motions in lindhard s planar continuum model over long time intervals that grow with the particle energy our averaging method includes a generalization of a method of besjes to quasiperiodic functions with truncated diophantine conditions which allows us to exclude resonances due to low order axial directions although our methods are mathematically rigorous we believe the results are physically realistic notably the estimates of the critical energy and of the initial directions for planar channeling

pricing on electricity market based on coupled continuous time random walk concept abstract in this paper we propose a model of electricity market based on the forward rate dynamics described by a diffusion with jumps as a generalization of the classical diffusion approach we consider jump components resulting from a coupled continuous time random walk ctrw with jump lengths proportional to the corresponding inter jump time intervals in the framework of the model we derive a formula for the euro price of a standard european call option showing applicability of ctrw processes for pricing of financial instruments the result obtained by an advance theory of semimartingales is an essential extension of the pricing formula derived in the classical diffusion model of the forward rate dynamics it indicates an influence of both the continuous and the jump parts of the forward rate process on the option price

fractional brownian motion time changed by gamma and inverse gamma process abstract many real time series exhibit behavior adequate to long range dependent data additionally very often these time series have constant time periods and also have characteristics similar to gaussian processes although they are not gaussian therefore there is need to consider new classes of systems to model these kinds of empirical behavior motivated by this fact in this paper we analyze two processes which exhibit long range dependence property and have additional interesting characteristics which may be observed in real phenomena both of them are constructed as the superposition of fractional brownian motion fbm and other process in the first case the internal process which plays role of the time is the gamma process while in the second case the internal process is its inverse we present in detail their main properties paying main attention to the long range dependence property moreover we show how to simulate these processes and estimate their parameters we propose to use a novel method based on rescaled modified cumulative distribution function for estimation of parameters of the second considered process this method is very useful in description of rounded data like waiting times of subordinated processes delayed by inverse subordinators by using the monte carlo method we show the effectiveness of proposed estimation procedures finally we present the applications of proposed models to real time series

fokker planck type equations associated with fractional brownian motion controlled by infinitely divisible processes abstract in this paper we study the anomalous diffusion process driven by fractional brownian motion delayed by general infinitely divisible subordinator we show the analyzed process is the stochastic representation of the fokker planck type equation that describes the probability density function of an introduced model moreover we study main characteristics of the examined process the first two moments that allow us in special cases for classification of it as a system with accelerating subdiffusion property

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

non fickian flux for advection dispersion with immobile periods the fractal mobile immobile model mim is intermediate between advection dispersion ade and fractal fokker planck ffkpe equations it involves two time derivatives whose orders are 1 and between 0 and 1 on the left hand side whereas all mentioned equations have identical right hand sides the fractal mim model accounts for non fickian effects that occur when tracers spread in media because of through flow and can get trapped by immobile sites the solid matrix of a porous material may contain such sites so that non fickian spread is actually observed within the context of the fractal mim model we present a mapping that allows the computation of fluxes on the basis of the density of spreading particles the mapping behaves as fickian flux at early times and tends to a fractional derivative at late times by means of this mapping we recast the fractal mim model into conservative form which is suitable to deal with sources and bounded domains mathematical proofs are illustrated by comparing the discretized fractal p d e with monte carlo simulations

anomalous diffusion in stochastic systems with nonhomogeneously distributed traps the stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed the particle is subjected to a stochastic stimulation obeying a general levy stable statistics and experiences long rests due to nonhomogeneously distributed traps the memory is taken into account by subordination of that process to a random time then the subordination equation is position dependent the problem is approximated by a decoupling of the medium structure and memory and exactly solved for a power law position dependence of the memory in the case of the gaussian statistics the density distribution and moments are derived depending on geometry and memory parameters the system may reveal both the subdiffusion and enhanced diffusion the similar analysis is performed for the levy flights where the finiteness of the variance follows from a variable noise intensity near a boundary two diffusion regimes are found in the bulk and near the surface the anomalous diffusion exponent as a function of the system parameters is derived

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

the adiabatic limit of wave map flow on a two torus the s2 valued wave map flow on a lorentzian domain r where is any flat two torus is studied the cauchy problem with initial data tangent to the moduli space of holomorphic maps s2 is onsidered in the limit of small initial velocity it is proved that wave maps in this limit converge in a precise sense to geodesics in the moduli space of holomorphic maps with respect to the l2 metric this establishes in a rigorous setting a long standing informal conjecture of ward

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

explicit representations for multiscale levy processes and asymptotics of multifractal conservation laws nonlinear conservation laws driven by levy processes have solutions which in the case of supercritical nonlinearities have an asymptotic behavior dictated by the solutions of the linearized equations thus the explicit representation of the latter is of interest in the nonlinear theory in this paper we concentrate on the case where the driving levy process is a multiscale stable anomalous diffusion which corresponds to the case of multifractal conservation laws considered in biler et al j differ equations 148 9 1998 stud math 135 231 1999 ann inst henri poincare anal nonlineaire 18 613 2001 and stud math 148 171 2001 the explicit representations building on the previous work on single scale problems see e g gorska and penson phys rev e 83 061125 2011 are developed in terms of the special functions such as meijer g functions and are amenable to direct numerical evaluations of relevant probabilities

exact moduli space metrics for hyperbolic vortex polygons exact metrics on some totally geodesic submanifolds of the moduli space of static hyperbolic n vortices are derived these submanifolds denoted as n m are spaces of cn invariant vortex configurations with n single vortices at the vertices of a regular polygon and m n n coincident vortices at the polygon s center the geometric properties of n m are investigated and it is found that n n 1 is isometric to the hyperbolic plane of curvature 3 n 1 the geodesic flow on n m and a geometrically natural variant of geodesic flow recently proposed by collie and tong the dynamics of chern simons vortices phys rev d part fields gravit cosmol 78 065013 2008 e print arxiv hep th 0805 0602 are analyzed in detail

the kahler potential of abelian higgs vortices we calculate the kahler potential for the samols metric on the moduli space of abelian higgs vortices on r2 in two different ways the first uses a scaling argument the second depends on a variant of the relationship between accessory parameters and the regularized action in liouville field theory the kahler potential on the moduli space of vortices on h2 is also derived and we are led to a geometrical reinterpretation of these vortices

asymptotic behavior of the supremum tail probability for anomalous diffusions abstract in this paper we investigate asymptotic behavior of the tail probability for subordinated self similar processes with regularly varying tail probability we show that the tail probability of the one dimensional distributions and the supremum tail probability are regularly varying with the pre factor depending on the moments of the subordinating process we can apply our result to the so called anomalous diffusion

feature extraction of handwritten japanese characters by spine functions for relaxation matching abstract this paper proposes a method of extracting features of handwritten japanese characters by using spline approximation this method represents a character by contours expressed by well approximating functions and stable breakpoints which characterize the connection of the strokes so that it provides proper features for handwritten japanese character recognition with relaxation matching its effectiveness tested on a japanese character database is also shown

governing equations and solutions of anomalous random walk limits fachbereich mathematik university of dortmund 44221 dortmund germany received 5 june 2002 published 30 december 2002 continuous time random walks model anomalous diffusion coupling allows the magnitude of particlejumps to depend on the waiting time between jumps governing equations for the long time scaling limits ofthese models are found to have fractional powers of coupled space and time differential operators explicitsolutions and scaling properties are presented for these equations which can be used to model ow in porousmedia and other physical systems doi 10 1103 physreve 66 060102 pacs number s 05 40 2a 02 50 cw 05 60 2k 47 55 mhi introduction

fractional oscillator driven by a gaussian noise abstract the time dependent response is studied of a fractional oscillator driven by a gaussian white noise applying the method of characteristic functionals combined with the technique of integral transforms we derive explicit formulas for the variances of position and velocity and the asymptotic power spectrum of the oscillator the effects of the order of fractional derivative on statistical characteristics of the oscillator are analyzed numerically

diffusion in hamiltonian systems with a small stochastic perturbation abstract we study some examples of hamiltonian systems perturbed by a small random noise which are relevant in accelerator physics generalization to other hamiltonian systems is briefly sketched starting from the liouville equation we derive a fokker planck equation for the distribution function in the unperturbed action angle variables which is valid for a vanishingly small noise when the angle distribution has relaxed we write a simple equation for the distribution in the action however there is evidence that such an equation governs the distribution averaged on the angle even before the relaxation occurs suggesting that an averaging principle does apply we compare the solution of this equation with the numerical simulation obtained by using a symplectic integrator of the stochastic hamilton s equations we consider also stochastically perturbed isochronous hamiltonian and the corresponding area preserving map for the map we write the action diffusion coefficients and compare with the numerical simulations the averaged distribution functions both in the action obtained by solving the diffusion equation

fractional vector calculus for fractional advection dispersion abstract we develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient divergence and curl and a fractional divergence theorem and stokes theorem these basic tools are then applied to provide a physical explanation for the fractional advection dispersion equation for flow in heterogeneous porous media

a stochastic theory of adiabatic invariance leti be a set of invariants for a system of differential equations with an ordero e vector field when order e perturbations of zero mean are added to the system we show that under suitable regularity and ergodicity conditions i becomes an adiabatic invariant with maximal variations of order one on time scales of order 1 e2 in the stochastically perturbed case i behaves asymptotically for small e like a diffusion process on 1 e2 time scales the results also apply to an interesting class of deterministic perturbations this study extends the results of khas minskii on stochastically averaged systems as well as some of the deterministic methods of averaging to such invariants

low energy dynamics of a cp1 lump on the sphere low energy dynamics in the unit charge sector of the cp1 model on spherical space space time s2 r is treated in the approximation of geodesic motion on the moduli space of static solutions a six dimensional manifold with nontrivial topology and metric the structure of the induced metric is restricted by consideration of the isometry group inherited from global symmetries of the full field theory evaluation of the metric is then reduced to finding five functions of one coordinate which may be done explicitly some totally geodesic submanifolds are found and the qualitative features of motion on these described

exact solutions of triple order time fractional differential equations for anomalous relaxation and diffusion i the accelerating case a b s t r a c t in recent years the interest around the study of anomalous relaxation and diffusion processes is increased due to their importance in several natural phenomena moreover a further generalization has been developed by introducing time fractional differentiation of distributed order which ranges between 0 and 1 we refer to accelerating processes when the driving power law has a changing in time exponent whose modulus tends from less than 1 to 1 and to decelerating processes when such an exponent modulus decreases in time moving away from the linear behaviour accelerating processes are modelled by a time fractional derivative in the riemann liouville sense while decelerating processes by a time fractional derivative in the caputo sense here the focus is on the accelerating case while the decelerating one is considered in the companion paper after a short reminder about the derivation of the fundamental solution for a general distribution of time derivative orders we consider in detail the triple order case for both accelerating relaxation and accelerating diffusion processes and the exact results are derived in terms of an infinite series of h functions the method adopted is new and it makes use of certain properties of the generalized mittag leffler function and the h function moreover it provides an elegant generalization of the method introduced by langlands 2006 t a m langlands physica a 367 2006 136 to study the double order case of accelerating diffusion processes

boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

memory effects and macroscopic manifestation of randomness it is shown that due to memory effects the complex behavior of components in a stochastic system can be transmitted to macroscopic evolution of the system as a whole within the markov approximation widely used in ordinary statistical mechanics memory effects are neglected as a result a time scale separation between the macroscopic and the microscopic level of description exists the macroscopic differential picture is not a consequence of microscopic nondifferentiable dynamics on the other hand the presence of complete memory in a system means that all its components have the same behavior if the memory function has no characteristic time scales the correct description of the macroscopic evolution of such systems has to be in terms of the fractional calculus

diffusion and memory effects for stochastic processes and fractional langevin equations we consider the diffusion processes defined by stochastic differential equations when the noise is correlated a functional method based on the dyson expansion for the evolution operator associated to the stochastic continuity equation is proposed to obtain the fokker planck equation after averaging over the stochastic process in the white noise limit the standard result corresponding to the stratonovich interpretation of the non linear langevin equation is recovered when the noise is correlated the averaged operator series cannot be summed unless a family of time dependent operators commutes in the case of a linear equation the constraints are easily worked out the process defined by a linear langevin equation with additive noise is gaussian and the probability density function of its fluctuating component satisfies a fokker planck equation with a time dependent diffusion coefficient the same result holds for a linear langevin equation with a fractional time derivative defined according to caputo 1 in the generic linear or nonlinear case approximate equations for small noise amplitude are obtained for small correlation time the evolution equations further simplify in agreement with some previous alternative derivations the results are illustrated by the linear oscillator with coloured noise and the fractional wiener process where the numerical simulation for the probability density and its moments is compared with the analytical solution

black scholes formula in subdiffusive regime in the classical approach the price of an asset is described by the celebrated black scholes model in this paper we consider a generalization of this model which captures the subdiffusive characteristics of financial markets we introduce a subdiffusive geometric brownian motion as a model of asset prices exhibiting subdiffusive dynamics we find the corresponding fractional fokker planck equation governing the dynamics of the probability density function of the introduced process we prove that the considered model is arbitrage free and incomplete we find the corresponding subdiffusive black scholes formula for the fair prices of european options and show how these prices can be evaluated using monte carlo methods we compare the obtained results with the classical ones

tempered fractional calculus fractional derivatives and integrals are convolutions with a power law multiplying by an exponential factor leads to tempered fractional derivatives and integrals tempered fractional diffusion equations where the usual second derivative in space is replaced by a tempered fractional derivative govern the limits of random walk models with an exponentially tempered power law jump distribution the limiting tempered stable probability densities exhibit semi heavy tails which are commonly observed in finance tempered power law waiting times lead to tempered fractional time derivatives which have proven useful in geophysics the tempered fractional derivative or integral of a brownian motion called a tempered fractional brownian motion can exhibit semi long range dependence the increments of this process called tempered fractional gaussian noise provide a useful new stochastic model for wind speed data a tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations and it also provides a useful new correlation model in time series

sigma model solitons and their moduli space metrics we discuss the supersymmetric model and its soliton solutions in 2 1 dimensions we classify supersymmetric maps and derive bogomolny bounds we also give the modified superalgebra and describe the metric on the parameter space of solitons

a new thinning algorithm for gray scale images by the relaxation technique abstract a new thinning algorithm for gray scale images based on the relaxation technique is proposed thinning is treated as a process of assigning each picture point to either a skeleton or a nonskeleton category so that relaxation can be used to perform thinning contextual information existing in input data is utilized in the thinning process so that line straightness can be preserved five classes are created to classify each point with four classes belonging to the skeleton category for different orientations and one to the nonskeleton category an initial probability value for assigning a point to each class is computed first in the relaxation process interactions take place among the probability values of neighbouring points making the skeleton probability values of the points in compatible orientations to enhance one another the probability value for each point to belong to the nonskeleton class of each point is also refined according to the point type the relaxation process is terminated when it converges to a condition in which points lying on skeletons achieve high skeleton probability values while other points have unity nonskeleton probability values experimental results show that the proposed approach produces good thinning results in keeping line straightness

experimental evidence of the role of compound counting processes in random walk approaches to fractional dynamics we present dielectric spectroscopy data obtained for gallium doped cd 0 99 mn 0 01 te ga mixed crystals which exhibit a very special case of the two power law relaxation pattern with the high frequency power law exponent equal to 1 we explain this behavior which cannot be fitted by any of the well known empirical relaxation functions in a subordinated diffusive framework we propose a diffusion scenario based on a renormalized clustering of a random number of spatio temporal steps in the continuous time random walk such a construction substitutes the renewal counting process which is used in the classical continuous time random walk methodology with a compound counting one as a result we obtain an appropriate relaxation function governing the observed nonstandard pattern and we show the importance of the compound counting processes in studying fractional dynamics of complex systems

models of anomalous diffusion the subdiffusive case abstract the paper discusses a model for anomalous diffusion processes their one point probability density functions p d f are exact solutions of fractional diffusion equations the model reflects the asymptotic behavior of a jump anomalous random walk process with random jump sizes and random inter jump time intervals with infinite means and variances which do not satisfy the law of large numbers in the case when these intervals have a fractional exponential p d f the fractional komogorov feller equation for the corresponding anomalous diffusion is provided and methods of finding its solutions are discussed finally some statistical properties of solutions of the related langevin equation are studied the subdiffusive case is explored in detail the emphasis is on a rigorous presentation which however would be accessible to the physical sciences audience

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

investigation of cumulative growth process via fibonacci method and fractional calculus in this study cumulative growth of a physical quantity with fibonacci method and fractional calculus is handled the development of the growth process is described in terms of fibonacci numbers mittag leffler and exponential functions a compound growth process with the contribution of a constant quantity is also discussed for the accumulation of residual quantity equilibrium and lessening cases are discussed to the best of our knowledge compound growth process is solved for the first time in the framework of fractional calculus in this sense differintegral order of fractional calculus has been achieved a physical content it is emphasized that in the basis of qualification of the fractional calculus for describing genuine complex physical systems with respect to ordinary descriptions is the cumulative growth mechanism with fibonacci method it is concluded that compound diminution and growth process mechanisms can be taken as a basis for the comprehension of derivative and integral operations in fractional calculus

lump dynamics in the ℂ p 1 model on the torus the topology and geometry of the moduli space m 2 of degree 2 static solutions of the p 1 model on a torus spacetime t 2 are studied it is proved that m 2 is homeomorphic to the left coset space g g 0 where g is a certain eight dimensional noncompact lie group and g 0 is a discrete subgroup of order 4 low energy two lump dynamics is approximated by geodesic motion on m 2 with respect to a metric g defined by the restriction to m 2 of the kinetic energy functional of the model this lump dynamics decouples into a trivial centre of mass motion and nontrivial relative motion on a reduced moduli space it is proved that m 2 g is geodesically incomplete and has only finite diameter a low dimensional geodesic submanifold is identified and a full description of its geodesics obtained

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

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

anomalous diffusion without scale invariance asymptotic behaviour of a new class of anomalous diffusion equations for subdiffusive transport defined in terms of generalized distributed fractional order time derivatives is considered the effect of slowly varying factors on the scaling function of asymptotic solutions is demonstrated the origin of slowly varying scaling factors in the ctrw models is discussed

the evaluation of geometric asian power options under time changed mixed fractional brownian motion abstract in this paper the geometric asian option pricing problem is investigated under the assumption that the underlying stock price is assumed following a mixed fractional subdiffusive black scholes model and the geometric average asian option pricing formula is derived under this assumption we then apply the results to value asian power options on the stocks that pay constant dividends when the payoff is a power function finally lower bound of asian options and some special cases are provided

dispersionless limit of the noncommutative potential kp hierarchy and solutions of the pseudodual chiral model in 2 1 dimensions the usual dispersionless limit of the kp hierarchy does not work in the case where the dependent variable has values in a noncommutative e g matrix algebra passing over to the potential kp hierarchy there is a corresponding scaling limit in the noncommutative case which turns out to be the hierarchy of a pseudodual chiral model in 2 1 dimensions pseudodual to a hierarchy extending ward s modified integrable chiral model applying the scaling procedure to a method generating exact solutions of a matrix potential kp hierarchy from solutions of a matrix linear heat hierarchy leads to a corresponding method that generates exact solutions of the matrix dispersionless potential kp hierarchy i e the pseudodual chiral model hierarchy we use this result to construct classes of exact solutions of the su m pseudodual chiral model in 2 1 dimensions including various multiple lump configurations

stability and lack of memory of the returns of the hang seng index abstract in this paper we show that the logarithmic returns of the hang seng index from january 2 1987 to november 14 2005 statistically resemble a sequence of independent identically distributed levy stable random variables this is in stark contrast to xiu and jin 2007 39 where long memory farima processes with gaussian noise were suggested as well fitted to the data the lack of memory is checked by using lo s modified r s statistic and a new method of estimation of the memory parameter d which applies the notion of empirical mean squared displacement in order to test stability of the data we employ several statistical tests based on the empirical distribution function finally we also show that the returns possess no conditional heteroscedasticity property thus excluding the arch garch family of processes as possible underlying models

time changed geometric fractional brownian motion and option pricing with transaction costs abstract this paper deals with the problem of discrete time option pricing by a fractional subdiffusive black scholes model the price of the underlying stock follows a time changed geometric fractional brownian motion by a mean self financing delta hedging argument the pricing formula for the european call option in discrete time setting is obtained

anomalous diffusion schemes underlying the cole cole relaxation the role of the inverse time α stable subordinator abstract the paper presents the random variable formalism of the anomalous diffusion processes the emphasis is on a rigorous presentation of asymptotic behaviour of random walk processes with infinite mean random time intervals between jumps we elucidate the role of the so called inverse time stochastic process the main mathematical tool that allows us to modify the dynamics of standard relaxation processes and give rise to the nonexponential decay of modes in particular we show that the brownian motion in combination with an appropriate inverse time process may lead not only to exponential but also to the nonexponential relaxation responses

some recent advances in theory and simulation of fractional diffusion processes to offer an insight into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of current interest i the well scaled passage to the limit from continuous time random walk under power law assumptions to space time fractional diffusion ii the asymptotic universality of the mittag leffler waiting time law in time fractional processes iii our method of parametric subordination for generating particle trajectories

stability of classical solutions of two dimensional grassmannian models we show that the only finite action solutions of the two dimensional grassmannian model that are stable under small fluctuations are the anti instanton solutions

fractional fokker planck equation with space and time dependent drift and diffusion in this paper we aim to answer the question proposed by magdziarz stoch proc appl 119 3238 3252 2009 i e we investigate the solution of an anomalous diffusion equation with time and space dependent force and diffusion coefficient first we try to find the stochastic representation of this equation which means the pdf of this stochastic process is rightly the solution of the equation we aim to solve then we also simulate the sample paths of the stochastic process at last taking advantage of the stochastic representation method we employed monte carlo method to approximate the solution of the mentioned equation

tempered stable levy motion driven by stable subordinator a b s t r a c t in this article we propose a new model for financial data description combining two independent mechanisms namely the tempered stable process and inverse stable subordinator we obtain a new model which captures not only the tempered stable character of the underlying data but also such a property as periods in which the values of an asset stay on the same level moreover we classify our system to the family of subdiffusive processes and investigate its tail behavior we describe in detail testing and estimation procedures for the proposed model in the last step we calibrate our model to the real data

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

thinning approach for noisy digital patterns abstract many problems difficult to solve by traditional computer algorithms can be explored by the principles of human visual perception from the hint of human visual symmetry a thinning approach for noisy digital patterns is presented in this paper it includes three main algorithms algorithm 1 is designed to find the effective circular range for each pixel containing the maximum effective information algorithm 2 is designed to compute the symmetry score of the pixel distribution in the circular range and convert the symmetry information into a gray scale image algorithm 3 provides a modified gray scale thinning algorithm for thinning the gray scale image to obtain a connected thin line experiments show that the thinning results obtained by the proposed method are properly acceptable whensnr 2 2 db and the connectedness of the thin line can be preserved

multiscale conservation laws driven by levy stable and linnik diffusions asymptotics shock creation preservation and dissolution recent work has shown that the solutions of the fractal conservation laws driven by levy alpha stable diffusions exhibit shocks for bounded odd and convex on the positive half line initial data when the parameter alpha 1 we study the analogous situation for the levy alpha linnik diffusions in which case the local behavior is strikingly different although we are able to establish analytically that the large time behavior of the two types of conservation laws are similar but the main new insights obtained via large scale numerical experiments is that for any 0 alpha le 2 the conservation laws driven by alpha linnik diffusions display shocks that do not dissipate over time while those for alpha stable diffusion 0 alpha le 1 do we formulate rigorous conjectures based on these numerical experiments

fractional diffusion model for force distribution in static granular media we present the results of a numerical and an experimental investigation of the probability distribution of normal contact forces in static packs of particles with two different hardnesses force distributions are computed and compared with existing models and experimental data it is found that the probability distribution function of normal contact forces p f is well described by a semiempirical model derived from a fractional diffusion equation this model reproduces most of the features common to force distributions observed in experimental and numerical studies including the finite value for p f as the forces tend to zero the results indicate that the fractional model fits well both the numerical and experimental data over a wide range of particle deformations in contrast to the existing models these results provide an insight into the physics of granular media and complement previous findings

integrated stationary ornstein uhlenbeck process and double integral processes abstract we find a representation of the integral of the stationary ornstein uhlenbeck isou process in terms of brownian motion b t moreover we show that under certain conditions on the functions f and g the double integral process dip d t t g s s f u d b u d s can be thought as the integral of a suitable gauss markov process some theoretical and application details are given among them we provide a simulation formula based on that representation by which sample paths probability densities and first passage times of the isou process are obtained the first passage times of the dip are also studied

investigation of the cumulative diminution process using the fibonacci method and fractional calculus abstract in this study we investigate the cumulative diminution phenomenon for a physical quantity and a diminution process with a constant acquisition quantity in each step in a viscous medium we analyze the existence of a dynamical mechanism that underlies the success of fractional calculus compared with standard mathematics for describing stochastic processes by proposing a fibonacci approach where we assume that the complex processes evolves cumulatively in fractal space and discrete time thus when the differential integral order is attained this indicates the involvement of the viscosity of the medium in the evolving process the future value of the diminishing physical quantity is obtained in terms of the mittag leffler function mlf and two rheological laws are inferred from the asymptotic limits thus we conclude that the differential integral calculus of fractional mathematics implicitly embodies the cumulative diminution mechanism that occurs in a viscous medium

black scholes model under subordination abstract in this paper we consider a new mathematical extension of the black scholes bs model in which the stochastic time and stock share price evolution is described by two independent random processes the parent process is brownian and the directing process is inverse to the totally skewed strictly stable process the subordinated process represents the brownian motion indexed by an independent continuous and increasing process this allows us to introduce the long term memory effects in the classical bs model

correlated continuous time random walk and option pricing in this paper we study a correlated continuous time random walk cctrw with averaged waiting time whose probability density function pdf is proved to follow stretched gaussian distribution then we apply this process into option pricing problem supposing the price of the underlying is driven by this cctrw we find this model captures the subdiffusive characteristic of financial markets by using the mean self financing hedging strategy we obtain the closed form pricing formulas for a european option with and without transaction costs respectively at last comparing the obtained model with the classical black scholes model we find the price obtained in this paper is higher than that obtained from the black scholes model a empirical analysis is also introduced to confirm the obtained results can fit the real data well

some experiments in relaxation image matching using corner features abstract a relaxation method based on patterns of local features is used to find matches between pairs of images or subimages that differ in position or orientation a local operator is applied to the two images to detect two sets of corners c1 cm and d1 dn each of them characterized by position orientation contrast and sharpness of the angle for each pair ci dj a figure of merit is computed and a relaxation process is used to iteratively adjust these figures of merit based on the merits of other pairs in approximately corresponding positions after a few iterations of this process good matches pairs having much better merit than their next best choices are clustered yielding sets of transformation parameters shift vectors or rotation angles under which many corners correspond this method has yielded good results for tv images of objects such as tools and industrial parts as well as for aerial images of terrain

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

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

generalized langevin equation with fractional derivative and long time correlation function we investigate the motion of a particle governed by a generalized langevin equation with fractional derivative nonlocal dissipative force and long time correlation function we derive general expressions for the variances with a linear external force we also analyze their asymptotic behaviors for the power law correlation function without external force

kramers escape problem for fractional klein kramers equation with tempered α stable waiting times in this paper we extend the subdiffusive klein kramers model in which the waiting times are modeled by the stable laws to the case of waiting times belonging to the class of tempered stable distributions we introduce a generalized version of the klein kramers equation in which the fractional riemman liouville derivative is replaced with a more general integro differential operator this allows a transition from the initial subdiffusive character of motion to the standard diffusion for long times to be modeled taking advantage of the corresponding langevin equation we study some properties of the tempered dynamics in particular we approximate solutions of the tempered klein kramers equation via monte carlo methods also we study the distribution of the escape time from the potential well and compare it to the classical results in the kramers escape theory finally we derive the analytical formula for the first passage time distribution for the case of free particles we show that the well known sparre andersen scaling holds also for the tempered subdiffusion

relaxation matching techniques a comparison many different relaxation schemes have been proposed for image analysis tasks we have developed a general matching procedure for comparing semantic network descriptions of images and we have implemented a variety of relaxation techniques an automatic segmentation and description system is used to produce the image representations so that the matching procedures must cope with variations in feature values missing objects and possible multiple matches this environment is used to test different relaxation matching schemes under a variety of conditions the best performance of those we compared in terms of the number of iterations and the number of errors is for the gradient based optimization approach of faugeras and price the related optimization approach of hummel and zucker performed almost as well with differences primarily in difficult matches i e where much of the evidence is against the match for instance poor segmentations the product combination rule proposed by peleg was extremely fast indeed too fast to work when global context is needed the classical rosenfeld hummel and zucker method is included for historical comparisons and performed only adequately producing fewer correct matches and taking more iterations

non markovian diffusion equations and processes analysis and simulations abstract in this paper we introduce and analyze a class of diffusion type equations related to certain non markovian stochastic processes we start from the forward drift equation which is made non local in time by the introduction of a suitable chosen memory kernel k t the resulting non markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l t we then consider the subordinated process y t x l t where x t is a markovian diffusion the corresponding time evolution of the marginal density function of y t is governed by a non markovian fokker planck equation which involves the memory kernel k t we develop several applications and derive the exact solutions we consider different stochastic models for the given equations providing path simulations

the geodesic approximation for lump dynamics and coercivity of the hessian for harmonic maps the most fruitful approach to studying low energy soliton dynamics in field theories of bogomol nyi type is the geodesic approximation of manton in the case of vortices and monopoles stuart has obtained rigorous estimates of the errors in this approximation and hence proved that it is valid in the low speed regime his method employs energy estimates which rely on a key coercivity property of the hessian of the energy functional of the theory under consideration in this article we prove an analogous coercivity property for the hessian of the energy functional of a general sigma model with compact kahler domain and target we go on to prove a continuity property for our result and show that for the cp1 model on s2 the hessian fails to be globally coercive in the degree 1 sector we present numerical evidence which suggests that the hessian is globally coercive in a certain equivariance class of the degree n sector for n 2 we also prove that within the geodesic approximation a single cp1 lump movi

exact moments in a continuous time random walk with complete memory of its history we present a continuous time generalization of a random walk with complete memory of its history phys rev e 70 045101 r 2004 and derive exact expressions for the first four moments of the distribution of displacement when the number of steps is poisson distributed we analyze the asymptotic behavior of the normalized third and fourth cumulants and identify new transitions in a parameter regime where the random walk exhibits superdiffusion these transitions which are also present in the discrete time case arise from the memory of the process and are not reproduced by fokker planck approximations to the evolution equation of this random walk

the higher order heat type equations via signed levy stable and generalized airy functions we study the higher order heat type equation with first time and mth spatial partial derivatives m 2 3 we demonstrate that its exact solutions for m even can be constructed with the help of signed levy stable functions for m odd the same role is played by a special generalization of the airy ai function that we introduce and study this permits one to generate the exact and explicit heat kernels pertaining to these equations we examine analytically and graphically the spatial and temporary evolution of particular solutions for simple initial conditions

fully parallel thinning with tolerance to boundary noise abstract a new fully parallel thinning algorithm is developed and evaluated in this paper to solve the noise spurs problem and preserve geometric properties efficiently the algorithm not only prevents the excessive erosions but also lessens the creation of spurious end points for an image with boundary noise when two input images are similar in shape but with boundary noise our skeletons produced appear more consistent in topology as compared to those using other algorithms although a few additional neighbors other than 3 3 are considered in the deletability conditions the smoothing procedure prior to thinning is avoided the parallel thinning algorithm runs very fast and can be implemented in real time several english and chinese characters and the difficult patterns often illustrated in the literature are also experimented to show the efficiency and consistency of our algorithm

on fractional tempered stable processes and their governing differential equations we derive the governing equation of the tempered stable subordinator hereafter tss which generalizes the space fractional differential equation satisfied by the law of the stable subordinator itself this equation is expressed in terms of the shifted fractional derivative of order 0 1 coinciding with the stability parameter we then generalize this equation by introducing a time fractional derivative of order 0 1 resp 1 1 and we prove that it is satisfied by the law of a tss time changed by the inverse of a stable subordinator resp by the stable subordinator itself the corresponding processes can therefore be called fractional ts processes finally we provide fractional extensions of the relativistic stable processes which we define as a brownian motion with a random time argument represented by independent fractional ts processes of order resp 1

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