the stokes phenomenon in the confluence of the hypergeometric equation using riccati equation abstract in this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one we study the consequence of the divergence of solutions at the irregular singular point for the unfolded system our study covers a full neighborhood of the origin in the confluence parameter space in particular we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system for this study we consider values of the confluence parameter taken in two sectors covering the complex plane in each sector we study the monodromy of a first integral of a riccati system related to the hypergeometric equation then on each sector we include the presence of logarithmic terms into a continuous phenomenon and view a stokes multiplier related to a 1 summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators one at each singular point to coincide

on the laplace transform of the frechet distribution we calculate exactly the laplace transform of the frechet distribution in the form x 1 exp x 0 0 x for arbitrary rational values of the shape parameter i e for l k with l k 1 2 the method employs the inverse mellin transform the closed form expressions are obtained in terms of meijer g functions and their graphical illustrations are provided a rescaled frechet distribution serves as a kernel of frechet integral transform it turns out that the frechet transform of one sided levy law reproduces the frechet distribution

lie properties of symmetric elements in group rings abstract let be an involution of a group g extended linearly to the group algebra kg we prove that if g contains no 2 elements and k is a field of characteristic p 2 then the symmetric elements of kg are lie nilpotent lie n engel if and only if kg is lie nilpotent lie n engel

the computation of the cohomology rings of all groups of order 128 abstract we describe the computation of the mod 2 cohomology rings of all 2328 groups of order 128 one consequence is that all groups of order less than 256 satisfy the strong form of benson s regularity conjecture

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

bimodality of the interspike interval distributions for subordinated diffusion models of integrate and fire neurons abstract a subordinated langevin process with a random operational time in the form of an inverse strictly increasing levy type subordinator is considered as a generalization of the conventional perfect and leaky integrate and fire neuron models the parent process is given by standard brownian motion the effect of the random activity of synaptic inputs which arises from other neurons forming local and distant networks is modeled via a levy exponent of the subordinator using a first passage time formulation in an external force field we find exact expressions for the laplace transform of the output interspike interval isi density more detailed analysis is presented on the properties of the isi distribution in the case of the levy exponent which corresponds to the truncated double order time fractional diffusion equation for the probability density of the membrane potential particularly it is shown that at some parameter regimes the isi density exhibits a bimodal structure moreover it is demonstrated that the isis regularity is maximized at an intermediate value of the mean input current

group identities on symmetric units abstract let f be an infinite field of characteristic different from 2 g a group and an involution of g extended by linearity to an involution of the group algebra fg here we completely characterize the torsion groups g for which the symmetric units of fg satisfy a group identity when is the classical involution induced from g g 1 g g this result was obtained in a giambruno s k sehgal a valenti symmetric units and group identities manuscripta math 96 1998 443 461

universal microscopic correlation functions for products of truncated unitary matrices we investigate the spectral properties of the product of m complex non hermitian random matrices that are obtained by removing l rows and columns of larger unitary random matrices uniformly distributed on the group u n l such matrices are called truncated unitary matrices or random contractions we first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed n l and m given by a standard determinantal point process in the complex plane the weight however is non standard and can be expressed in terms of the meijer g function the explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large n and l limits at fixed m at strong non unitarity with l n finite the eigenvalues condense on a domain inside the unit circle at the edge and in the bulk we find the same universal microscopic kernel as for a single complex non hermitian matrix from the ginibre ensemble at the origin we find the same new universality classes labeled by m as for the product of m matrices from the ginibre ensemble keeping a fixed size of truncation l when n goes to infinity leads to weak non unitarity with most eigenvalues on the unit circle as for unitary matrices here we find a new microscopic edge kernel that generalizes the known results for m 1 we briefly comment on the case when each product matrix results from a truncation of different size lj

n dimensional fractional fokker planck equation and its solutions for anomalous radial two phase flow in porous media in this paper we present a time fractional fokker planck equation ffpe for radial two phase flow of liquid and gas in porous media the ffpe of order a is solved for both two and three dimensional flow patterns using the laplace transform method the general solutions of the ffpe for both two and three dimensional flows are given as a convolution integral of the input and a kernel in the laplace domain special solutions for a large value and a periodic boundary condition are also given in the time domain when the inverse laplace transform can be found analytically the ffpe for two phase flow in porous media presented in this paper is the first report of its kind

finite groups of symplectic automorphisms of k3 surfaces in positive characteristic we show that mukai s classication of nite groups which may act symplectically on a complex k3 surface extends to positive characteristic p under assumptions that i the order of the group is coprime to p and ii either the surface or its quotient is not birationally isomorphic to a supersingular k3 surface with artin invariant 1 in the case without assumption ii we classify all possible new groups which may appear we prove that assumption i on the order of the group is always satised if p 11 for p 2 3 5 11 we give examples of k3 surfaces with nite symplectic automorphism groups of order divisible by p which are not contained in mukai s list

cm fields with relative class number one we will show that the normal cm fields with relative class number one are of degrees 216 moreover if we assume the generalized riemann hypothesis then the normal cm fields with relative class number one are of degrees 96 and the cm fields with class number one are of degrees 104 by many authors all normal cm fields of degrees 96 with class number one are known except for the possible fields of degree 64 or 96 consequently the class number one problem for normal cm fields is solved under the generalized riemann hypothesis except for these two cases

new generalized coherent states we construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation de q x dx e q qx with 0 q 1 and e q 0 1 we use this function in place of the usual exponential to generate new coherent states q z from the vacuum which are normalized and continuous in their label z these states allow the resolution of unity and a corresponding weight function is furnished by the exact solution of the associated stieltjes moment problem they also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally ordered function of boson operators we exemplify this by showing that the photon number statistics for these states is sub poissonian for any q 1 the states q z are squeezed we obtain and discuss their signal to quantum noise ratio the function e q x allows a natural generation of multiboson coherent states of arbitrary multiplicity which is impossible for the usual coherent states for q 1 all the above resul

group rings of finite strongly monomial groups central units and primitive idempotents abstract we compute the rank of the group of central units in the integral group ring z g of a finite strongly monomial group g the formula obtained is in terms of the strong shoda pairs of g next we construct a virtual basis of the group of central units of z g for a class of groups g properly contained in the finite strongly monomial groups furthermore for another class of groups g inside the finite strongly monomial groups we give an explicit construction of a complete set of orthogonal primitive idempotents of q g finally we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of z g this for metacyclic groups g of the form g c q m c p n with p and q different primes and the cyclic group c p n of order p n acting faithfully on the cyclic group c q m of order q m

continuous time random walk with correlated waiting times based on the langevin description of the continuous time random walk ctrw we consider a generalization of ctrw in which the waiting times between the subsequent jumps are correlated we discuss the cases of exponential and slowly decaying persistent power law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time in the case of exponential type correlations the sub diffusion at short times is slower than in the absence of correlations at long times the behavior of the mean squared displacement is the same as in uncorrelated ctrw for power law correlations we find subdiffusion characterized by the same exponent at all times which appears to be smaller than the one in uncorrelated ctrw interestingly in the limiting case of an extremely long power law correlations the sub diffusion exponent does not tend to zero but is bounded from below by the subdiffusion exponent corresponding to a short time behavior in the case of exponential correlations

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

presentations for unit groups of modular group algebras of groups of order 16 for a p group g and the field f of p elements let v denote the group of normalized units of the group algebra fg generators and relations are provided for v for each group g of order dividing 16 the presentations are sufficiently concise to permit transcription for machine calculation with v some applications are described a theoretical method for obtaining presentations for v is developed it is most effective when p 2 or when p 3 and the commutator subgroup g is of order 3 implementation involves extensive calculation in fg

spectral decomposition of nonlinear systems with memory we present an alternative approach to the analysis of nonlinear systems with long term memory that is based on the koopman operator and a levy transformation in time memory effects are considered to be the result of interactions between a system and its surrounding environment the analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale on average the time evolution of a mode follows a mittag leffler function and the system can be described using the fractional calculus the general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation when analyzing data from an ill defined black box system the spectral decomposition in terms of mittag leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods consequently the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators or operators with long term memory in time when model details are unknown

on a family of symmetric hypergeometric functions of several variables and their euler type integral representation abstract this paper is devoted to the family g n of hypergeometric series of any finite number of variables the coefficients being the square of the multinomial coefficients l 1 l n l 1 l n where n z 1 all these series belong to the family of the general appell lauricella s series it is shown that each function g n can be expressed by an integral involving the previous one g n 1 thus this family can be represented by a multidimensional euler type integral what suggests some explicit link with the gelfand kapranov zelevinsky s theory of a hypergeometric systems or with the aomoto s theory of hypergeometric functions the quasi invariance of each function g n with regard to the action of a finite number of involutions of c n is also established finally a particular attention is reserved to the study of the functions g 2 and g 3 each of which is proved to be algebraic or to be expressed by the legendre s elliptic function of the first kind

asymptotic solutions of decoupled continuous time random walks with superheavy tailed waiting time and heavy tailed jump length distributions we study the long time behavior of decoupled continuous time random walks characterized by superheavy tailed distributions of waiting times and symmetric heavy tailed distributions of jump lengths our main quantity of interest is the limiting probability density of the position of the walker multiplied by a scaling function of time we show that the probability density of the scaled walker position converges in the long time limit to a nondegenerate one only if the scaling function behaves in a certain way this function as well as the limiting probability density are determined in explicit form also we express the limiting probability density which has heavy tails in terms of the fox h function and find its behavior for small and large distances

closed form solutions for the levy stable distribution the levy stable distribution is the attractor of distributions which hold power laws with infinite variance this distribution has been used in a variety of research areas for example in economics it is used to model financial market fluctuations and in statistical mechanics it is used as a numerical solution of fractional kinetic equations of anomalous transport this function does not have an explicit expression and no uniform solution has been proposed yet this paper presents a uniform analytical approximation for the levy stable distribution based on matching power series expansions for this solution the trans stable function is defined as an auxiliary function which removes the numerical issues of the calculations of the levy stable distribution then the uniform solution is proposed as a result of an asymptotic matching between two types of approximations called the inner solution and the outer solution finally the results of analytical approximation are compared to the numerical results of the levy stable distribution function making this uniform solution valid to be applied as an analytical approximation

on the finite subgroups of u 3 of order smaller than 512 we use the smallgroups library to find the finite subgroups of u 3 of order smaller than 512 which possess a faithful three dimensional irreducible representation from the resulting list of groups we extract those groups that cannot be written as direct products with cyclic groups these groups are the basic building blocks for models based on finite subgroups of u 3 all resulting finite subgroups of su 3 can be identified using the well known list of finite subgroups of su 3 derived by miller blichfeldt and dickson at the beginning of the 20th century furthermore we prove a theorem which allows us to construct infinite series of finite subgroups of u 3 from a special type of finite subgroups of u 3 this theorem is used to construct some new series of finite subgroups of u 3 the first members of these series can be found in the derived list of finite subgroups of u 3 of order smaller than 512 in the last part of this work we analyze some interesting finite subgroups of u 3 especially the group which is closely related to the important su 3 subgroup s4

first passage time distributions of anomalous biased diffusion with double absorbing barriers abstract we investigate the first passage time fpt problem of anomalous diffusion governed by the galilei variant fractional diffusion advection equation in the semi infinite and finite domains subject to an absorbing boundary condition we obtain explicit solutions for the fpt distributions and the corresponding laplace transforms for both zero and constant drift cases by using the method of separation of variables as well as the properties of the fox h function an important relation between the fpt distributions corresponding to one and two absorbing barriers is revealed to determine the conditional fpt distributions it shows that the proportion between the conditional fpt distributions only depends on the general peclet number we further discuss the asymptotic behavior of the fpt distributions and confirm our theoretical analysis by numerical results

the groups of order 256 abstract building on earlier work a new method for generating descriptions of p groups is developed the theory and implementation of this method are described and its application in determining the 56 092 groups of order 256 is outlined

universality of stretched gaussian asymptotic behaviour for the fractional fokker planck equation in external force fields we introduce a heterogeneous fractional fokker planck equation hffpe on heterogeneous fractal structure media describing systems involving external force fields the hffpe is shown to obey the generalized einstein relation and its stationary solution is the boltzmann distribution it is proven that the asymptotic shape of its solution is a stretched gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for constant and generic potentials with polar singularity at origin

operator solutions for fractional fokker planck equations we obtain exact results for fractional equations of fokker planck type using the evolution operator method we employ exact forms of one sided levy stable distributions to generate a set of self reproducing solutions explicit cases are reported and studied for various fractional order of derivatives different initial conditions and for different versions of fokker planck operators

nonspectral relaxation in one dimensional ornstein uhlenbeck processes the relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates which are typically considered to be independent of the initial condition the rates follow from the spectrum of a hermitian operator obtained by a similarity transformation of the initial fokker planck operator however some initial conditions are mapped by this similarity transformation to functions which growat infinity these cannot be expanded in terms of the eigenfunctions of a hermitian operator and show different relaxation patterns considering the exactly solvable examples of gaussian and generalized levy ornstein uhlenbeck processes oups we show that the relaxation rates belong to the hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise while for an ordinary oup initial conditions leading to nonspectral relaxation can be considered exotic for generalized oups driven by levy noise such initial conditions are the rule doi 10 1103 physrevlett 110 150602

critical dynamics in systems controlled by fractional kinetic equations abstract the field theory renormalization group is used for analyzing the fractional langevin equation with the order of the temporal derivative 0 1 fractional laplacian of the order and gaussian noise correlator the case of non linearity m with odd m 3 is considered it is proved that the model is multiplicatively renormalizable propagators were found in the momentum and coordinate representation expressed in terms of fox s h functions existence of the dissipative scaling regime in the framework of the e expansion for 2 1 l l 1 2 is proved requirement of the continuous dependence of the critical exponents on imposes the condition m 3 the main quantitative result is the calculation of the dynamical critical exponent z for 1 2 up to e 2 we have obtained for it the expression z 1 2 4 0 1555 e 2 o e 3

simple components and central units in group algebras abstract let g be a finite group berman dokl akad nauk 106 1956 767 and witt j reine angew math 190 1952 231 evaluate independently the number of simple components of the group algebra fg when f is a field of characteristic 0 in this paper we extend this result to fields of arbitrary characteristic which does not divide the order of g we also compute the rank of the group of the central units of z g and obtain an alternative proof of a well known result of ritter and sehgal

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

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

the groups of order 128 abstract a computer assisted determination of the 2328 groups of order 128 and of their 115 isoclinism families is described previous work by rodemich on these groups is corrected the provision of access to the group descriptions obtained is discussed

efficient numerical solution of the time fractional diffusion equation by mapping from its brownian counterpart the solution of a caputo time fractional diffusion equation of order 0 1 is expressed in terms of the solution of a corresponding integer order diffusion equation we demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem in the context of an n point finite difference time discretisation the mapping allows for an improvement in time computational complexity from o n 2 to o n given a precomputation of o n 1 ln n the mapping is applied successfully to the least squares fitting of a fractional advection diffusion model for the current in a time of flight experiment resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes a mapping between fractional diffusion and normal diffusion is explored numerically an efficient solution method for fractional diffusion equations is proposed a benchmark fractional order model fit for scher montroll transport is presented

some classes of semisimple group and loop algebras over finite fields abstract we determine the structure of the semisimple group algebra of certain groups over the rationals and over those finite fields where the wedderburn decompositions have the least number of simple components we apply our work to obtain similar information about the loop algebras of indecomposable ra loops and to produce negative answers to the isomorphism problem over various fields

noether s problem for groups of order 32 abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed field noether s problem asks under what situations the fixed field k g will be rational purely transcendental over k theorem let g be a finite group of order 32 with exponent e if char k 2 or k is any field containing a primitive eth root of unity then k g is rational over k

algebraic a hypergeometric functions we formulate and prove a combinatorial criterion to decide if an a hypergeometric system of differential equations has a full set of algebraic solutions or not this criterion generalises the so called interlacing criterion in the case of hypergeometric functions of one variable

on the first passage of one sided levy motions abstract we study the first passage problem for one sided levy motions how does such a motion started at the origin cross a barrier positioned at the point x x 0 since one sided levy motions are pure jump processes they always leap over barriers rather than crossing them continuously we hence explore the following issues i first passage times fpts how long would it take the motion to cross the barrier ii first passage leapovers fpls how far would the motion leap over the barrier and iii what is dependence between the fpts and the fpls formulae for the joint laplace transforms of the fpts and fpls are derived and the scaling limits of the fpts and fpls are computed the scaling limits turn out to display dramatically different behavior when i the underlying levy motion has finite cumulants and when ii the underlying levy motion is heavy tailed special attention is devoted to the investigation of the first passage problem for self similar one sided levy motions

eigenvalue statistics for product complex wishart matrices the eigenvalue statistics for complex wishart matrices where is equal to the product of r complex gaussian matrices and the inverse of s complex gaussian matrices are considered in the case r s the exact form of the global density is computed the averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function for finite n the eigenvalue probability density function is computed and is shown to be an example of a biorthogonal ensemble a double contour integral form of the corresponding correlation kernel is derived which allows the hard edge scaled limit to be computed the limiting kernel is given in terms of certain meijer g functions and is identical to that found in the recent work of kuijlaars and zhang in the case s 0 properties of the kernel and corresponding correlation functions are discussed

groups and nilpotent lie rings whose order is the sixth power of a prime abstract we prove that there are 3p2 39p 344 24gcd p 1 3 11gcd p 1 4 2gcd p 1 5 isomorphism types of groups and nilpotent lie rings with order p6 for every prime p 5 we establish the result and power commutator presentations for the groups in various ways the most novel method constructs product presentations for nilpotent lie rings with order p6 and then uses the baker campbell hausdorff formula to construct power commutator presentations for the corresponding groups public access to the group presentations is provided via a database distributed with computer algebra systems

groups with two extreme character degrees and their normal subgroups we study the finite groups g for which the set cd g of irreducible complex character degrees consists of the two most extreme possible values that is 1 and g z g 1 2 we are easily reduced to finite p groups for which we derive the following group theoretical characterization they are the p groups such that ig z g l is a square and whose only normal subgroups are those containing g or contained in z g by analogy we also deal with pgroups such that g z g i p2n 1 is not a square and we prove that cd g 1 pn if and only if a similar property holds for any n g either g n or nz g z g p the proof of these results requires a detailed analysis of the structure of the p groups with any of the conditions above on normal subgroups which is interesting for its own sake it is especially remarkable that these groups have small nilpotency class and that if the nilpotency class is greater than 2 then the index of the centre is small and in some cases we may even bound the order of g

exact solution of a generalized kramers fokker planck equation retaining retardation effects in order to describe non gaussian kinetics in weakly damped systems the concept of continuous time random walks is extended to particles with finite inertia one thus obtains a generalized kramers fokker planck equation which retains retardation effects i e nonlocal couplings in time and space it is shown that despite this complexity exact solutions of this equation can be given in terms of superpositions of gaussian distributions with varying variances in particular the long time behavior of the respective low order moments is calculated

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 langevin equation overdamped underdamped and critical behaviors the dynamical phase diagram of the fractional langevin equation is investigated for a harmonically bound particle it is shown that critical exponents mark dynamical transitions in the behavior of the system four different critical exponents are found i ensuremath alpha c 0 402 ifmmode pm else textpm fi 0 002 marks a transition to a nonmonotonic underdamped phase ii ensuremath alpha r 0 441 dots marks a transition to a resonance phase when an external oscillating field drives the system and iii ensuremath alpha ensuremath chi 1 0 527 dots and iv ensuremath alpha ensuremath chi 2 0 707 dots mark transitions to a double peak phase of the loss when such an oscillating field present as a physical explanation we present a cage effect where the medium induces an elastic type of friction phase diagrams describing over and underdamped regimes with or without resonances show behaviors different from normal

computation of galois groups over function fields symmetric function theory provides a basis for computing galois groups which is largely independent of the coefficient ring an exact algorithm has been implemented over q t 1 t 2 t m in maple for degree up to 8 a table of polynomials realizing each transitive permutation group of degree 8 as a galois group over the rationals is included

fundamental solution of the tempered fractional diffusion equation in this paper we consider the space time fractional diffusion equation dt u x t k dx u x t 0 x r t 0 with the tempered riemann liouville derivative of order 0 1 in space and the caputo derivative of order 0 1 in time the fundamental solution which turns out to be a spatial probability density function is given in computable series form as well as in integral representation the spatial moments of the probability density function are determined explicitly for an arbitrary order n 0 moreover green s function of the untempered neutral fractional diffusion equation is analyzed in view of absolute and relative extreme points at the end of this article we point out a remarkably and important integral representation for accurate evaluation of the m wright mainardi function m x of order 0 1 and arguments x r0

divergent series and memory of the initial condition in the long time solution of some anomalous diffusion problems we consider various anomalous d dimensional diffusion problems in the presence of an absorbing boundary with radial symmetry the motion of particles is described by a fractional diffusion equation their mean square displacement is given by r 2 t 0 1 resulting in normal diffusive motion if 1 and subdiffusive motion otherwise for the subdiffusive case in sufficiently high dimensions divergent series appear when the concentration or survival probabilities are evaluated via the method of separation of variables while the solution for normal diffusion problems is at most divergent as t 0 the emergence of such series in the long time domain is a specific feature of subdiffusion problems we present a method to regularize such series and in some cases validate the procedure by using alternative techniques laplace transform method and numerical simulations in the normal diffusion case we find that the signature of the initial condition on the approach to the steady state rapidly fades away and the solution approaches a single the main decay mode in the long time regime in remarkable contrast long time memory of the initial condition is present in the subdiffusive case as the spatial part 1 r describing the long time decay of the solution to the steady state is determined by a weighted superposition of all spatial modes characteristic of the normal diffusion problem the weight being dependent on the initial condition interestingly 1 r turns out to be independent of the anomalous diffusion exponent

subdiffusion in a bounded domain with a partially absorbing reflecting boundary the exit time of a subdiffusive process from a bounded domain with a partially absorbing reflecting boundary is considered the short time and long time behaviors of the exit time probability density are investigated by using a spectral decomposition on the basis of the laplace operator eigenfunctions rotation invariant domains are analyzed in depth in order to illustrate the use of theoretical formulas and to compare them to numerical simulations the asymptotic results obtained are relevant for describing subdiffusion inside a living cell with a semipermeable membrane in a chemical reactor filled with catalytic grains of finite reactivity or in mineral or biological samples which are probed by nuclear magnetic resonance measurements subject to surface relaxation