simulation of the continuous time random walk of the space fractional diffusion equations in this article we discuss the solution of the space fractional diffusion equation with and without central linear drift in the fourier domain and show the strong connection between it and the a stable levy distribution 0 a 2 we use some relevant transformations of the independent variables x and t to find the solution of the space fractional diffusion equation with central linear drift which is a special form of the space fractional fokker planck equation which is useful in studying the dynamic behaviour of stochastic differential equations driven by the non gaussian levy noises we simulate the continuous time random walk of these models by using the monte carlo method

fractional laplacian time space models for linear and nonlinear lossy media exhibiting arbitrary frequency power law dependency frequency dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2 the standard time domain partial differential equation models can describe merely two extreme cases of frequency independent and frequency squared dependent attenuations the otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation in this study a linear integro differential equation wave model was developed for the anomalous attenuation by using the space fractional laplacian operation and the strategy is then extended to the nonlinear burgers equation a new definition of the fractional laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional laplacian under the szabo s smallness approximation where attenuation is assumed to be much smaller than the wave number the linear model is found consistent with arbitrary frequency power law dependency

representation of random walk in fractal space time abstract to analyze the anomalous diffusion on a fractal structure with fractal in the time axis we propose a statistical representation given by a path integral method in arbitrary fractal space time using the method we can understand easily several properties of the non gaussian type behavior and a differential equation for the path integral is derived finally to check the validity of this theory analytical results in this paper are applied to the random walk on the two dimensional sierpinski carpet which agree precisely with numerical results by monte carlo simulations in the paper of fujiwara and yonezawa phys rev e 51 1995 2277

diffusion on multifractals abstract we introduce a class of random fields with variable mean square regularity order defined on multifractal domains the local singularity order of these random fields then depends on the initial variable mean square regularity order and on the variable local singularity exponent of the multifractal measure defining the local dimension of the domain considered the theory is developed in a generalized framework through the covariance factorization using the tools of reproducing kernel hilbert spaces and fractional sobolev spaces of variable order

robust satisfiability of systems of equations we study the problem of robust satisfiability of systems of nonlinear equations namely whether for a given continuous function f k rn on a finite simplicial complex k and 0 it holds that each function g k rn such that g f has a root in k via a reduction to the extension problem of maps into a sphere we particularly show that this problem is decidable in polynomial time for every fixed n assuming dim k 2n 3 this is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis via a reverse reduction we prove that the problem is undecidable when dim k 2n 2 where the threshold comes from the stable range in homotopy theory for the lucidity of our exposition we focus on the setting when f is simplexwise linear such functions can approximate general continuous functions and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings

extremal functions and best constants to an inequality involving hardy potential and critical sobolev exponent abstract in this paper we study the asymptotic behavior of radial extremal functions to an inequality involving hardy potential and critical sobolev exponent based on the asymptotic behavior at the origin and the infinity we shall deduce a strict inequality between two best constants finally as an application of this strict inequality we consider the existence of a nontrivial solution of a quasilinear brezis nirenberg type problem with hardy potential and critical sobolev exponent

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

embeddability in r 3 is np hard we prove that the problem of deciding whether a two or three dimensional simplicial complex embeds into r3 is np hard our construction also shows that deciding whether a 3 manifold with boundary tori admits an s3 filling is np hard the former stands in contrast with the lower dimensional cases which can be solved in linear time and the latter with a variety of computational problems in 3 manifold topology for example unknot or 3 sphere recognition which are in np co np membership of the latter problem in co np assumes the generalized riemann hypotheses our reduction encodes a satisfiability instance into the embeddability problem of a 3 manifold with boundary tori and relies extensively on techniques from low dimensional topology most importantly dehn fillings of manifolds with boundary tori

existence of weak solutions to degenerate p laplacian equations and integral formulas abstract we study the problem of solving some general integral formulas and then apply the conclusions to obtain results about the existence of weak solutions of various degenerate p laplacian equations we adapt variational calculus methods and the mountain pass lemma without the palais smale condition and we use an abstract version of lions concentration compactness principle ii

existence results for brezis nirenberg problems with hardy potential and singular coefficients abstract in this paper we consider the existence of non trivial solutions to semi linear brezis nirenberg type problems with hardy potential and singular coefficients first we shall study the corresponding eigenvalue problem and obtain some basic properties of eigenvalues and asymptotic estimates of the eigenfunctions and approximating eigenfunctions secondly we consider the extremal functions of the best embedding constant and get some crucial estimates for the cut off function of the extremal functions thirdly applying different variational theorems for distinct cases of those parameters appearing in the equation we obtain two existence results for non trivial solutions to semi linear brezis nirenberg type problems our existence results are divided into non resonant and resonant cases

on the existence and multiplicity of solutions for a class of singular elliptic problems abstract in this paper we prove the existence and multiplicity of solutions for a class of elliptic problems with singular coefficients we also prove the existence of a sign changing solution for a brezis nirenberg type problem with hardy potential and singular coefficients our proofs are based on the krasnoselskii genus the concentration compactness principle of lions and the linking theorem

the fractional fick s law for non local transport processes fick s law is extensively adopted as a model for standard diffusion processes however requiring separation of scales it is not suitable for describing non local transport processes we discuss a generalized non local fick s law derived from the space fractional diffusion equation generating the levy feller statistics this means that the fundamental solutions can be interpreted as levy stable probability densities in the feller parameterization with index 1 2 and skewness 2 we explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities flux and gradient it turns out that the presence of asymmetry 0 plays a fundamental role it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter gradient transport

multiple solutions for the brezis nirenberg problem with a hardy potential and singular coefficients by energy estimates and by establishing a local ps condition we establish the multiplicity of solutions to a class of semi linear brezis nirenberg type problems with a hardy term and singular coefficients via the pseudo index theory

critical growth problems for polyharmonic operators we prove that critical growth problems for polyharmonic operators admit nontrivial solutions for a wide class of lower order perturbations of the critical term the results highlight the phenomenon of bifurcation of the critical dimensions discovered by pucci and serrin moreover we show that another bifurcation seems to appear for nonresonant dimensions

on a linearity between fractal dimension and order of fractional calculus in holder space abstract in this paper the linear relationship between fractal dimensions and the order of fractional calculus of functions in holder space has been mainly investigated under specific holder condition the linear connection between box dimension and the order of riemann liouville fractional integral and derivative has been proved this linear connection is also established with k dimension and packing dimension some function examples have been given in the end

embeddability in the 3 sphere is decidable we show that the following algorithmic problem is decidable given a 2 dimensional simplicial complex can it be embedded topologically or equivalently piecewise linearly in r3 by a known reduction it suffices to decide the embeddability of a given triangulated 3 manifold x into the 3 sphere s3 the main step which allows us to simplify x and recurse is in proving that if x can be embedded in s3 then there is also an embedding in which x has a short meridian that is an essential curve in the boundary of x bounding a disk in s3 b x with length bounded by a computable function of the number of tetrahedra of x

polynomial time computation of homotopy groups and postnikov systems in fixed dimension for several computational problems in homotopy theory we obtain algorithms with running time polynomial in the input size in particular for every fixed k 2 there is a polynomial time algorithm that for a 1 connected topological space x given as a finite simplicial complex or more generally as a simplicial set with polynomial time homology computes the kth homotopy group pi k x as well as the first k stages of a postnikov system of x combined with results of an earlier paper this yields a polynomial time computation of x y i e all homotopy classes of continuous mappings x y under the assumption that y is k 1 connected and dim x y where y is k 1 connected and dim x y and the question is the extendability of f to all of x the algorithms are based on the notion of a simplicial set with polynomial time homology which is an enhancement of the notion of a simplicial set with effective homology developed earlier by sergeraert and his coworkers our polynomial time algorithms are obtained by showing that simplicial sets with polynomial time homology are closed under various operations most notably cartesian products twisted cartesian products and classifying space one of the key components is also polynomial time homology for the eilenberg maclane space k z 1 provided in another recent paper by krcal matousek and sergeraert

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

sumudu transform in fractal calculus abstract the c calculus includes functions on fractal sets which are not differentiable or integrable using ordinary calculus sumudu transforms have an important role in control engineering problems because of preserving units the scaling property of domains easy visualization and transforming linear differential equations to algebraic equations that can be easily solved analogues of the laplace and sumudu transforms in c calculus are defined and the corresponding theorems are proved the generalized laplace and sumudu transforms involve functions with totally disconnected fractal sets in the real line linear differential equations on cantor like sets are solved utilizing fractal sumudu transforms the results are summarized in tables and figures illustrative examples are solved to give more details

solutions for semilinear elliptic equations with critical exponents and hardy potential abstract in this paper we answer affirmatively an open problem cf theorem 4 in ferrero and gazzola j differential equations 177 2001 494 let 0 be an open bounded domain r n n 5 and assume that 0 n 2 2 2 n 2 n 2 then for all 0 there exists a nontrivial solution with critical level in the range 0 1 n s n 2 for the problem u u x 2 u u 2 2 u in u 0 on

universality classes for asymptotic behavior of relaxation processes in systems with dynamical disorder dynamical generalizations of stretched exponential the asymptotic behavior of multichannel parallel relaxation processes for systems with dynamical disorder is investigated in the limit of a very large number of channels an individual channel is characterized by a state vector x which due to dynamical disorder is a random function of time a limit of the thermodynamic type in the x space is introduced for which both the volume available and the average number of channels tend to infinity but the average volume density of channels remains constant scaling arguments combined with a stochastic renormalization group approach lead to the identification of two different types of universal behavior of the relaxation function corresponding to nonintermittent and intermittent fluctuations respectively for nonintermittent fluctuations a dynamical generalization of the static huber s relaxation equation is derived which depends only on the average functional density of channels w t d w t the channels being classified according to their different relaxation rates w w t which are random functions of time for intermittent fluctuations a more complicated relaxation equation is derived which in addition to the average density of channels w t d w t depends also on a positive fractal exponent h which characterizes the fluctuations of the density of channels the general theory is applied for constructing dynamical analogs of the stretched exponential relaxation function for nonintermittent fluctuations the type of relaxation is determined by the regression dynamics of the fluctuations of the relaxation rate if the regression process is fast and described by an exponential attenuation function then after an initial stretched exponential behavior the relaxation process slows down and it is not fully completed even in the limit of very large times for self similar regression obeying a negative power law the relaxation process is less sensitive to the influence of dynamical disorder both for small and large times the relaxation process is described by stretched exponentials with the same fractal exponent as for systems with static disorder for large times the efficiency of the relaxation process is also slowed down by fluctuations similar patterns are found for intermittent fluctuations with the difference that for very large times and a slow regression process a crossover from a stretched exponential to a self similar algebraic relaxation function occurs some implications of the results for the study of relaxation processes in

change in order of phase transitions on fractal lattices abstract we re examine a population model which exhibits a continuous absorbing phase transition belonging to directed percolation in 1d and a first order transition in 2d and above studying the model on sierpinski carpets of varying fractal dimensions we examine at what fractal dimension 1 d f 2 the change in order occurs as well as commenting on the order of the transitions we produce estimates for the critical points and for continuous transitions some critical exponents

the role of the fox wright functions in fractional sub diffusion of distributed order the fundamental solution of the fractional diffusion equation of distributed order in time usually adopted for modelling sub diffusion processes is obtained based on its mellin barnes integral representation such solution is proved to be related via a laplace type integral to the fox wright functions a series expansion is also provided in order to point out the distribution of time scales related to the distribution of the fractional orders the results of the time fractional diffusion equation of a single order are also recalled and then re obtained from the general theory