on relating two approaches to fractional calculus abstract in this paper two methods are discussed for defining fractional powers of the nth order ordinary differential expression l x a 1 dx a 2 d x a n dx a n 1 d d dx where each ai i 1 2 n 1 is a complex number and x is real and positive an important role is played by the number m i 1 n 1 a i n when m 0 spectral theory is used to obtain a suitable formula for l c while the case m 0 considered in a previous paper by a c mcbride proc london math soc 3 45 1982 519 546 is dealt with by expressing l in terms of erdelyi kober operators we give accounts of each approach and go on to consider how the results for m 0 can be obtained as limiting cases of the results for m 0

diffusion in heterogeneous media diffusion processes in heterogeneous media and biological systems in particular are riddled with the difficult theoretical issue of whether the true origin of anomalous behavior is renewal or memory or a special combination of the two accounting for the possible mixture of renewal and memory sources of subdiffusion is challenging from a computational point of view as well this problem is exacerbated by the limited number of techniques available for solving fractional diffusion equations with time dependent coefficients we propose an iterative scheme for solving fractional differential equations with time dependent coefficients that is based on a parametric expansion in the fractional index we demonstrate how this method can be used to predict the long time behavior of nonautonomous fractional differential equations by studying the anomalous diffusion process arising from a mixture of renewal and memory sources

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

fractional diffusions with time varying coefficients this paper is concerned with the fractionalized diffusion equations governing the law of the fractional brownian motion bh t we obtain solutions of these equations which are probability laws extending that of bh t our analysis is based on mcbride fractional operators generalizing the hyper bessel operators l and converting their fractional power l into erdelyi kober fractional integrals we study also probabilistic properties of the random variables whose distributions satisfy space time fractional equations involving caputo and riesz fractional derivatives some results emerging from the analysis of fractional equations with time varying coefficients have the form of distributions of time changed random variables

characterizations and simulations of a class of stochastic processes to model anomalous diffusion in this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusions this class called generalized grey brownian motion ggbm is made up of self similar with stationary increments processes h sssi and depends on two real parameters 0 2 and 0 1 it includes fractional brownian motion when 0 2 and 1 and time fractional diffusion stochastic processes when 0 1 the latter have a marginal probability density function governed by timefractional diffusion equations of order the ggbm is defined through the explicit construction of the underlying probability space however in this paper we show that it is possible to define it in an unspecified probability space for this purpose we write down explicitly all the finite dimensional probability density functions moreover we provide different ggbm characterizations the role of them wright function which is related to the fundamental solution of the time fractional diffusion equation emerges as a natural generalization of the gaussian distribution furthermore we show that the ggbm can be represented in terms of the product of a random variable which is related to the m wright function and an independent fractional brownian motion this representation highlights the h sssi nature of the ggbm and provides a way to study and simulate the trajectories for this purpose we developed a random walk model based on a finite difference approximation of a partial integro differential equation of a fractional type

geometric isomorphism and minimum aberration for factorial designs with quantitative factors factorial designs have broad applications in agricultural engineering and scientific studies in constructing and studying properties of factorial designs traditional design theory treats all factors as nominal however this is not appropriate for experiments that involve quantitative factors for designs with quantitative factors level permutation of one or more factors in a design matrix could result in different geometric structures and thus different design properties in this paper indicator functions are introduced to represent factorial designs a polynomial form of indicator functions is used to characterize the geometric structure of those designs geometric isomorphism is defined for classifying designs with quantitative factors based on indicator functions a new aberration criteria is proposed and some minimum aberration designs are presented

brownian time processes the pde connection and the half derivative generator we introduce a class of interesting stochastic processes based on brownian time processes these are obtained by taking markov processes and replacing the time parameter with the modulus of brownian motion they generalize the iterated brownian motion ibm of burdzy and the markov snake of le gall and they introduce new interesting examples after defining brownian time processes we relate them to fourth order parabolic partial differential equations pde s we then study their exit problem as they exit nice domains in r d and connect it to elliptic pde s we show that these processes have the peculiar property that they solve fourth order parabolic pde s but their exit distribution at least in the standard brownian time process case solves the usual second order dirichlet problem we recover fourth order pde s in the elliptic setting by encoding the iterative nature of the brownian time process through its exit time in a standard brownian motion we also show that it is possible to assign a formal generator to these non markovian processes by giving such a generator in the half derivative sense

l kuramoto sivashinsky spdes vs time fractional spides exact continuity and gradient moduli 1 2 derivative criticality and laws abstract we establish exact dimension dependent spatio temporal uniform and local moduli of continuity for 1 the fourth order l kuramoto sivashinsky l ks spdes and for 2 the time fractional stochastic partial integro differential equations spides driven by the space time white noise in one to three dimensional spaces both classes were introduced with brownian time type kernel formulations by allouba in a series of articles starting in 2006 where he presented class 2 in its rigorous stochastic integral equations form he proved existence uniqueness and sharp spatio temporal holder regularity for the above two classes of equations in d 1 2 3 we show that both classes are 1 2 holder continuously differentiable in space when d 1 and we give the exact uniform and local moduli of continuity for the gradient in both cases this is unprecedented for spdes driven by the space time white noise our results on exact moduli show that the half derivative spide is a critical case it signals the onset of rougher modulus regularity in space than both time fractional spides with time derivatives of order 1 2 and l ks spdes this is despite the fact that they all have identical spatial holder regularity as shown earlier by allouba moreover we show that the temporal laws governing 1 and 2 are fundamentally different we relate l ks spdes to the houdre villa bifractional brownian motion yielding a chung type law of the iterated logarithm for these spdes we use the underlying explicit kernels and spectral harmonic analysis to prove our results on one hand this work builds on the recent works on delicate sample path properties of gaussian random fields on the other hand it builds on and complements allouba s earlier works on 1 and 2 similar regularity results hold for the allen cahn nonlinear members of 1 and 2 on compacts via change of measure

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

fractional diffusion equations and processes with randomly varying time in this paper the solutions u u x t to fractional diffusion equations of order 0 v 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes for the fractional equations of order v 1 2 n n 1 we show that the solutions u 1 2 n correspond to the distribution of the n times iterated brownian motion for these processes the distributions of the maximum and of the sojourn time are explicitly given the case of fractional equations of order v 2 3 n n 1 is also investigated and related to brownian motion and processes with densities expressed in terms of airy functions in the general case we show that u coincides with the distribution of brownian motion with random time or of different processes with a brownian time the interplay between the solutions u and stable distributions is also explored interesting cases involving the bilateral exponential distribution are obtained in the limit

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

geometrical finiteness for hyperbolic groups abstract in this paper we give an account of the notion of geometrical finiteness as applied to discrete groups acting on hyperbolic space of any dimension we prove the equivalence of various definitions of geometrical finiteness and describe the geometry of fundamental domains we give a complete account of when dirichiet domains are finite sided

subdiffusion with time dependent coefficients improved regularity and second order time stepping this article concerns second order time discretization of subdiffusion equations with time dependent diffusion coefficients high order differentiability and regularity estimates are established for subdiffusion equations with time dependent coefficients using these regularity results and a perturbation argument of freezing the diffusion coefficient we prove that the convolution quadrature generated by the second order backward differentiation formula with proper correction at the first time step can achieve second order convergence for both nonsmooth initial data and incompatible source term numerical experiments are consistent with the theoretical results

on almost fuchsian manifolds almost fuchsian manifold is a class of complete hyperbolic threemanifolds a such three manifold is a quasi fuchsian manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of 1 1 in such a manifold the minimal surface is unique and embedded hence one can parametrize these hyperbolic three manifolds by their minimal surfaces in this paper we obtain estimates on several geometric and analytical quantities of an almost fuchsian manifold m in terms of the data on the minimal surface in particular we obtain an upper bound for the hyperbolic volume of the convex core of m and an upper bound on the hausdorff dimension of the limit set associated to m we also constructed a quasi fuchsian manifold which admits more than one minimal surface and it does not admit a foliation of closed surfaces of constant mean curvature

fractional klein gordon equations and related stochastic processes this paper presents finite velocity random motions driven by fractional klein gordon equations of order alpha in 0 1 a key tool in the analysis is played by the mcbride s theory which converts fractional hyper bessel operators into erdelyi kober integral operators special attention is payed to the fractional telegraph process whose space dependent distribution solves a non homogeneous fractional klein gordon equation the distribution of the fractional telegraph process for alpha 1 coincides with that of the classical telegraph process and its driving equation converts into the homogeneous klein gordon equation fractional planar random motions at finite velocity are also investigated the corresponding distributions obtained as well as the explicit form of the governing equations fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations b n alpha with uniformly distributed orientation are considered the parameter n of b n alpha is itself a random variable with fractional poisson distribution so that fractionality acts as a subsampling of the changes of direction finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained extensions to n dimensional fractional random flights are envisaged as well as the fractional counterpart of the euler poisson darboux equation to which our theory applies

space time fractional diffusion on bounded domains abstract fractional diffusion equations replace the integer order derivatives in space and time by their fractional order analogues they are used in physics to model anomalous diffusion this paper develops strong solutions of space time fractional diffusion equations on bounded domains as well as probabilistic representations of these solutions which are useful for particle tracking codes

generalised minimum aberration construction results for symmetrical orthogonal arrays generalised minimum aberration is a recently established design criterion for the whole class of orthogonal arrays and fractional factorial designs the criterion is as its name suggests a generalisation of minimum aberration for regular designs and of minimum g 2 aberration for two level designs the aim of the criterion is to find designs which minimise in a certain sense the aliasing between main effects and interactions in this paper theoretical results are developed for finding symmetrical orthogonal arrays with generalised minimum aberration for more than two factor levels

lower bounds for lambda sb 1 on a finite volume hyperbolic manifold soit m une variete hyperbolique a n dimensions de volume fini v et soit 1 le premier element positif dans le spectre discret pour le probleme f f 0 sur m on considere des bornes inferieures pour 1 dans le cas de volume fini

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

generalized resolution for orthogonal arrays the generalized word length pattern of an orthogonal array allows a ranking of orthogonal arrays in terms of the generalized minimum aberration criterion xu and wu 2001 we provide a statistical interpretation for the number of shortest words of an orthogonal array in terms of sums of r values based on orthogonal coding or sums of squared canonical correlations based on arbitrary coding directly related to these results we derive two versions of generalized resolution for qualitative factors both of which are generalizations of the generalized resolution by deng and tang 1999 and tang and deng 1999 we provide a sufficient condition for one of these to attain its upper bound and we provide explicit upper bounds for two classes of symmetric designs factor wise generalized resolution values provide useful additional detail