exceptional units in a family of quartic number fields we determine all exceptional units among the elements of certain groups of units in quartic number fields these groups arise from a one parameter family of polynomials with two real roots

some exact results for the trapping of subdiffusive particles in one dimension we study a generalization of the standard trapping problem of random walk theory in which particles move subdiffusively on a one dimensional lattice we consider the cases in which the lattice is filled with a one sided and a two sided random distribution of static absorbing traps with concentration c the survival probability t that the random walker is not trapped by time t is obtained exactly in both versions of the problem through a fractional diffusion approach comparison with simulation results is made

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

random walk in chemical space of cantor dust as a paradigm of superdiffusion we point out that the chemical space of a totally disconnected cantor dust k n ensuremath subset e n is a compact metric space c n with the spectral dimension d s d ensuremath ell ngd where d and d ensuremath ell n are the fractal and chemical dimensions of k n respectively hence we can define a random walk in the chemical space as a markovian gaussian process the mapping of a random walk in c n into k n ensuremath subset e n defines the quenched l evy flight on the cantor dust with a single step duration independent of the step length the equations describing the superdiffusion and diffusion reaction front propagation ruled by the local quenched l evy flight on k n ensuremath subset e n are derived the use of these equations to model superdiffusive phenomena observed in some physical systems in which propagators decay faster than algebraically is discussed

fractional diffusion probability distributions and random walk models abstract we present a variety of models of random walk discrete in space and time suitable for simulating random variables whose probability density obeys a space time fractional diffusion equation

multidimensional solutions of space fractional diffusion equations green s functions and propagators for the multidimensional anisotropic spacetime fractional diffusion equation are obtained in integral form uniqueness of the solutions is studied in the framework

boundary problems for diffusion in a fluctuating potential abstract using a newly developed formalism adapted for diffusion in the presence of partially absorbing boundaries we treat the overdamped motion of a particle in a fluctuating potential and in a restricted space the potential is assumed to be linear with a slope switching between two possible values the switching process is taken as the standard markov alternating process we first construct the free dynamics propagator no boundaries which controls the evolution of the joint probability for both the particle s position and the state of the potential secondly at a given spatial point we affix a two channel boundary a given channel absorbs the trajectories arriving at the boundary with the potential in a given state we demonstrate the asymmetry of the filling process for the individual channels and we discuss their asymptotic occupation depending on the mean switching frequency finally we study the motion in a domain restricted by two boundaries of the above two channel type and the corresponding first passage time the simple problem setting allows for an explicit analysis of the correlation between the particle s motion and the potential switching process

the adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations in this paper we present the adomian decomposition method and its modifications combined with convergence acceleration techniques such as the diagonal pade approximants and the iterated shanks transforms to solve nonlinear fractional ordinary differential equations two nonlinear numeric examples demonstrate that either the diagonal pade approximants or the iterated shanks transforms can efficiently extend the effective convergence region of the decomposition series solution

hierarchical spatio temporal coupling in fractional wanderings i continuous time weierstrass flights abstract the one dimensional continuous time weierstrass flights ctwf model is considered in the framework of the nonseparable continuous time random walks formalism ctrw a novel spatio temporal coupling is introduced by assuming that in each scale the probability density for the flight and for waiting are joined hence we treat the spatio temporal relations in terms of the self similar structure of the weierstrass process this stochastic structure is characterized by the spatial fractional dimension 1 representing the flights and the temporal one 1 representing the waiting time was assumed here as the only independent truncation range in this work we study the asymptotic properties of the ctwf model for example by applying the method of steepest descents we obtained the particle propagator in the approximate scaling form p x t t pt 2 f where the scalingfunction f pt exp const pt while the scaling variable x t pt 2 is large the principal result of our analysis is that the exponents and depend on more fundamental ones and what leads to a novel scaling as a result of competition between exponents and an enhanced dispersive or normal diffusion was recognized in distinction from the prediction of the separable ctrw model where the enhanced diffusion is lost and the dispersive one is strongly limited it should be noted that we compare here partially thermalized versions of both approaches where some initial fluctuations were also included in agreement with the spirit of the theory of the renewal processes having the propogator we calculated for example the particle mean square displacement and found its novel asymptotic scaling with time for enhanced diffusion given by t1 2 1 in distinction from its diverging for

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

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

a family of cyclic quartic fields arising from modular curves we study a family of cyclic quartic fields arising from the covering of modular curves x1 16 xo 16 an integral basis and a fundamental system of units are found it is shown that a root of the quartic polynomial we construct is a translate of a cyclotomic period by an integer of the quadratic subfield of the quartic field recently 0 lecacheux 9 10 and h darmon 4 showed how to use coverings of modular curves to obtain cyclic extensions of q in particular they were able to give a geometric construction of a family of cyclic quintic fields discovered by e lehmer 11 the covering x1 n x0 n for n 2 has degree q n 2 and group z nz x i 1 for the quintic case they took n 25 which gave a cyclic covering of degree 10 then took the subcovering of degree 5 an important ingredient in the construction was the fact that xo 25 has genus 0 this also occurs for n 1 10 12 13 16 18 these all give trivial or quadratic coverings except for n 7 9 13 16 18 the values n 7 9 18 yield cubic extensions and can be shown to yield the family of polynomials x3 ax2 a 3 x 1 namely the simplest cubic fields 17 however it should be remarked that every cyclic cubic extension of q comes from a polynomial of this form if a is allowed to be rational similarly we are guaranteed that the quadratic extensions obtained from the coverings mentioned above correspond to polynomials of the form x2 ax 1 with a rational the case n 13 is treated by lecacheux 9 it might be suspected that the sextic fields she obtains are the same as the simplest sextics constructed by m n gras 6 however these latter fields were found by taking the fixed field of an element of order 6 in pgl2 q aut q x therefore they come from a covering of curves of genus 0 but x1 13 has genus 2 alternatively these sextic fields must be different because the discriminants of the quadratic and cubic subfields are different in the present paper we study the case n 16 as above it might be hoped that this case would give a geometric construction of the quartic fields studied received march 14 1990 revised october 22 1990 1980 mathematics subject classification 1985 revision primary 1 i r 16 research supported in part by n s f 1991 american mathematical society 0025 5718 91 1 00 25 per page

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

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

units in some families of algebraic number fields multi dimensional continued fractions associated with gl n z are introduced and applied to find systems of fundamental units in some families of totally real fields and fields with signature 2 1

hierarchical spatio temporal coupling in fractional wanderings ii diffusion phase diagram for weierstrass walks abstract the one dimensional weierstrass walks ww model is developed in the framework of the extended nonseparable continuous time random walk ctrw formalism 1 2 3 4 5 6 7 8 9 10 11 12 the ww model is a lacunary foundation of levy walks 6 7 8 9 10 11 12 generalized to a nonconstant velocity this nonconstant velocity is introduced by hierarchical coherent spatio temporal coupling adopted from the continuous time weierstrass flights ctwf model developed in the previous paper 13 hence for the probability density to pass by a walker in a single step a random displacement with finite velocity is constructed as a geometric series of the corresponding probability densities defined within a sequence of spatio temporal scales we calculated analytically and by monte carlo simulations the asymptotic in time mean square displacement msd of the walker obtaining very good agreement between both approaches also comparison with corresponding results of the ctwf model is made considering different types of the diffusion exponents we constructed a diffusion phase diagram on the plane defined by the spatial and temporal fractional dimensions which characterize our coupling we obtained a diffusion exponent as a function of these fractional dimensions covering all types of nonbiased diffusion known up to now from the dispersive one over the normal enhanced ballistic and hyperdiffusion up to the richardson law of diffusion which defines here a part of the borderline of the region where the msd diverges we observed that all kinds of anomalous diffusion are characterized by three types of diffusion exponents only for example we found an asymptotic scaling of msd to occur with time for enhanced diffusion which was discovered by us within the ctwf model but is valid for a much more extended range of spatio and temporal fractional dimensions

solutions of the initial value problem for nonlinear fractional ordinary differential equations by the rach adomian meyers modified decomposition method abstract in this paper we present the generalized adomian rach theorem and the generalized rach adomian meyers modified decomposition method for solving multi order nonlinear fractional ordinary differential equations we consider different classes of initial value problems for nonlinear fractional ordinary differential equations including the case of real valued orders and another case of rational valued orders which are solved by the present method this method can treat any analytic nonlinearity the coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme which does not involve integration operations compared with the classic adomian decomposition method

green function of the double fractional fokker planck equation path integral and stochastic differential equations the statistics of rare events the so called black swan events is governed by non gaussian distributions with heavy power like tails we calculate the green functions of the associated fokker planck equations and solve the related stochastic differential equations we also discuss the subject in the framework of path integration

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

dimensional influence on blow up in a superdiffusive medium thermal blow up in a superdiffusive medium with a localized energy source is examined for a spatial domain of infinite extent in one two and three dimensions an analysis of a nonlinear model of this problem reveals that the occurrence of a blow up depends upon the spatial dimension of the superdiffusive medium in one dimension a blow up always occurs in two and three dimensions a blow up can be avoided if the superdiffusive properties are sufficiently enhanced the asymptotic growth of the temperature near blow up is determined for energy sources whose output increases in either an algebraic or an exponential manner the results of the analysis have potential application to porous materials in which a pore filling agent is introduced to enhance the thermal transport properties

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

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

fractional diffusion exact representations of spectral functions for all the relevant transformed spaces i e fourier laplace and fourier laplace we present exact solutions of a fractional diffusion equation describing random transport on fractals the potential importance of such spectral representations lies in their applications to interpreting experimental measurements of anomalous diffusion processes in contrast to the well known asymptotic results the exact representations provide a much broader basis for comparison with data

representation of solutions and large time behavior for fully nonlocal diffusion equations abstract we study the cauchy problem for a nonlocal heat equation which is of fractional order both in space and time we prove four main theorems i a representation formula for classical solutions ii a quantitative decay rate at which the solution tends to the fundamental solution iii optimal l 2 decay of mild solutions in all dimensions iv l 2 decay of weak solutions via energy methods the first result relies on a delicate analysis of the definition of classical solutions after proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of ii next we use fourier analysis techniques to obtain the optimal decay rate for mild solutions here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions consequently the decay rate does not anymore improve when the dimension increases the theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis finally we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation our main idea is to show that the l 2 norm is actually a subsolution to a purely time fractional problem which allows us to use the known theory to obtain the result

invariance of a partial differential equation of fractional order under the lie group of scaling transformations abstract in this article a symmetry group of scaling transformations is determined for a partial differential equation of fractional order containing among particular cases the diffusion equation the wave equation and the fractional diffusion wave equation for its group invariant solutions an ordinary differential equation of fractional order with the new independent variable z xt 2 is derived the derivative then is an erdelyi kober derivative depending on a parameter its complete solution is given in terms of the wright and the generalized wright functions

diffusion equations for a markovian jumping process we consider a markovian jumping process which is defined in terms of the jump size distribution and the waiting time distribution with a position dependent frequency in the diffusion limit we assume the power law form for the frequency for small steps we derive the fokker planck equation and show the presence of the normal diffusion subdiffusion and superdiffusion for the l evy distribution of the step size we construct a fractional equation which possesses a variable coefficient and solve it in the diffusion limit then we calculate fractional moments and define the fractional diffusion coefficient as a natural extension to the cases with the divergent variance we also solve the master equation numerically and demonstrate that there are deviations from the l evy stable distribution for large wave numbers

stability of solutions of a class of nonlinear fractional laplacian parabolic problems abstract in this paper we study the forward and the backward in time problems for a class of nonlinear partial differential equations of fractional laplace operator with nonlinear source function the equations are generalized from lots of well known equations such as the burger equation the cahn hilliard equation herein we investigate the stability of the solution of the forward problem depended on the fractional order of the operator and the instability of the backward problem and then we establish a regularization method some numerical experiments are performed to verify efficiency and accuracy of our method

time and space fractional partial differential equations the fundamental solution for time and space fractional partial differential operator dt a2 2 0 is given in terms of the fox s h function here the time fractional derivative in the sense of generalized functions distributions dt is defined by the convolution dt f t t f t where t t 1 and f t 0 as t 0 and the fractional n dimensional laplace operator 2 is defined by its fourier transform with respect to spatial variable f 2g x f g x the solutions for initial value problems for time and space fractional partial differential equation in the sense of caputo and riemann liouville time fractional derivatives respectively are obtained by the fundamental solution

fox function representation of non debye relaxation processes applying the liouville riemann fractional calculus we derive and solve a fractional operator relaxation equation we demonstrate how the exponent of the asymptotic power law decay t relates to the order of the fractional operatordv dtv 0 1 continuous time random walk ctrw models offer a physical interpretation of fractional order equations and thus we point out a connection between a special type of ctrw and our fractional relaxation model exact analytical solutions of the fractional relaxation equation are obtained in terms of fox functions by using laplace and mellin transforms apart from fractional relaxation fox functions are further used to calculate fourier integrals of kohlrausch williams watts type relaxation functions because of its close connection to integral transforms the rich class of fox functions forms a suitable framework for discussing slow relaxation phenomena

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

the fractional schrodinger equation for delta potentials the fractional schrodinger equation is solved for the delta potential and the double delta potential for all energies the solutions are given in terms of fox s h function

heat conduction modeling by using fractional order derivatives the article deals with the heat conduction modeling a brief historical overview of the authors who have dealt with the heat conduction and overview of solving methods is listed in the introduction of article in the next section a mathematical model of one dimensional heat conduction with using derivatives of integer and fractional order is described the methods of solving models of heat conduction are described namely analytical and numerical methods in the case of numerical methods regards the finite difference method by using grunwald letnikov definition for the fractional time derivative implementation of these individual methods was realized in matlab the two libraries of m functions for the heat conduction model have been created namely heat conduction toolbox and fractional heat conduction toolbox at the conclusion of the article the simulations examples with using toolboxes are listed

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

existence and regularity of a linear nonlocal fokker planck equation with growing drift abstract the nonlocal fokker planck equations for a class of stochastic differential equations with non gaussian stable levy motion in euclidean space are studied the existence and uniqueness of weak solution are obtained with rough drift the solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded moreover the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity

balanced metrics on twisted higgs bundles a twisted higgs bundle on a kahler manifold x is a pair e phi consisting of a holomorphic vector bundle e and a holomorphic bundle morphism phi m otimes e rightarrow e for some holomorphic vector bundle m such objects were first considered by hitchin when x is a curve and m is the tangent bundle of x and also by simpson for higher dimensional base the hitchin kobayashi correspondence for such pairs states that e phi is polystable if and only if e admits a hermitian metric solving the hitchin equation this correspondence is a powerful tool to decide whether there exists a solution of the equation but it provides little information as to the actual solution in this paper we study a quantization of this problem that is expressed in terms of finite dimensional data and balanced metrics that give approximate solutions to the hitchin equation motivation for this study comes from work of donagi wijnholt jhep 05 068 2013 concerning balanced metrics for the vafa witten equations

second order accuracy finite difference methods for space fractional partial differential equations abstract space fractional partial differential equations are used for simulations of for example diffusion of radioactive materials and financial and other models which are characterized by heavy tailed distributions a number of first order accuracy finite difference methods have been proposed in the present paper we introduce second order accuracy finite difference methods with dirichlet boundary conditions these methods have a parameter in these schemes and the parameter stabilizes the schemes this means that there exist various schemes with second order accuracy but the stability of each scheme is different in the present paper we introduce the most stable scheme for any fractional calculus order by choosing the optimal parameter in addition we describe a phenomenon whereby the expected accuracy cannot be obtained if the analytical solution can be expanded to a series having less than second order around boundaries this also happens in both existing methods and the proposed methods in the present paper we develop the stability conditions for the proposed schemes and numerical examples of second order accuracy and accuracy decay are shown

solutions for a fractional nonlinear diffusion equation spatial time dependent diffusion coefficient and external forces we analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation we start our investigation by considering the linear case and the nonlinear case afterward the linear case is discussed taking fractional time and spatial derivatives into account in a unified approach we also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by d x t d t x for the nonlinear case we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation

approximate solutions of nonlinear fractional equations fractional exponential that are invariant under fractional derivatives elementary and special fractional functions are introduced approximate solutions to fractional burgers equation by using the homotopy perturbation method are obtained furthermore real integral representations for some h functions are found that may be very helpful in numerical computations

a finite difference method for fractional partial differential equation an implicit unconditional stable difference scheme is presented for a kind of linear space time fractional convection diffusion equation the equation is obtained from the classical integer order convection diffusion equations with fractional order derivatives for both space and time first order consistency unconditional stability and first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

persistent random walk of cells involving anomalous effects and random death the purpose of this paper is to implement a random death process into a persistent random walk model which produces sub ballistic superdiffusion levy walk we develop a stochastic two velocity jump model of cell motility for which the switching rate depends upon the time which the cell has spent moving in one direction it is assumed that the switching rate is a decreasing function of residence running time this assumption leads to the power law for the velocity switching time distribution this describes the anomalous persistence of cell motility the longer the cell moves in one direction the smaller the switching probability to another direction becomes we derive master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives we show that the random death of cells has an important implication for the transport process through tempering of the superdiffusive process in the long time limit we write stationary master equations in terms of exponentially truncated fractional derivatives in which the rate of death plays the role of tempering of a levy jump distribution we find the upper and lower bounds for the stationary profiles corresponding to the ballistic transport and diffusion with the death rate dependent diffusion coefficient monte carlo simulations confirm these bounds