implicit finite difference approximation for time fractional diffusion equations time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical brownian motion model in this paper we develop an implicit unconditionally stable numerical method to solve the one dimensional linear time fractional diffusion equation formulated with caputo s fractional derivative on a finite slab several numerical examples of interest are also included

some filter regularization methods for a backward heat conduction problem in this paper a one dimensional backward heat conduction problem is investigated it is well known that such problem is ill posed some filter regularization methods are used to solve it convergence estimates under two a priori bound assumptions for the exact solution are given based on the conditional stabilities finally numerical examples are given to show that our used numerical methods are effective and stable

a quasi reversibility regularization method for an inverse heat conduction problem without initial data in this paper we study an inverse heat conduction problem without initial data in a bounded domain in which the cauchy data at x 0 are given and the solution in 0

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

length expanding lipschitz maps on totally regular continua abstract the tent map is an elementary example of an interval map possessing many interesting properties such as dense periodicity exactness lipschitzness and a kind of length expansiveness it is often used in constructions of dynamical systems on the interval trees graphs the purpose of the present paper is to construct on totally regular continua i e on topologically rectifiable curves maps sharing some typical properties with the tent map these maps will be called length expanding lipschitz maps briefly lel maps we show that every totally regular continuum endowed with a suitable metric admits a lel map as an application we obtain that every non degenerate totally regular continuum admits an exactly devaney chaotic map with finite entropy and the specification property

a comparison of regularizations for an ill posed problem we consider numerical methods for a quasi boundary value regularization of the backward parabolic problem given by u t au 0 0 t t u t f where a is positive self adjoint and unbounded the regularization due to clark and oppenheimer perturbs the final value u t by adding u 0 where is a small parameter we show how this leads very naturally to a reformulation of the problem as a second kind fredholm integral equation which can be very easily approximated using methods previously developed by ames and epperson error estimates and examples are provided we also compare the regularization used here with that from ames and epperson

two regularization methods for a cauchy problem for the laplace equation abstract a cauchy problem for the laplace equation in a rectangle is considered cauchy data are given for y 0 and boundary data are for x 0 and x the solution for 0 y 1 is sought we propose two different regularization methods on the ill posed problem based on separation of variables both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates

the class breadth conjecture revisited abstract the class breadth conjecture for groups with prime power order was formulated by leedham green neumann and wiegold in 1969 we construct a new counter example to the conjecture it has order 2 19 and is a quotient of a 4 dimensional 2 uniserial space group we translate the conjecture to p uniserial space groups prove that these have finite cobreadth and provide an explicit upper bound we develop an algorithm to decide the conjecture for p uniserial space groups and use this to show that all 3 uniserial space groups of dimension at most 54 satisfy the conjecture we show that over every finite field there are lie algebras which fail the corresponding conjecture

fourier regularization for a backward heat equation abstract in this paper a simple and convenient new regularization method for solving backward heat equation fourier regularization method is given meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

determining surface temperatures from interior observations we consider the inverse heat transfer problem for the case of the one dimensional heat equation in a quarter plane and present a new solution algorithm together with error bounds of logarithmic convexity type our method is based on regularizing the initial value problem for the heat equation run sideways we describe a numerical experiment with rapidly varying temperature profiles and explain how the method can be extended to problems with temperature dependent thermal properties

calculus of variations in mean and convex lagrangians si est l operateur symplectique standard dans r n r n si h c 2 r n r n r alors les orbites quasi periodiques du systeme hamiltonien sr z h z sont exactement les points critiques de la fonctionnelle j definie par j z 7b m z z 2 h z

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

a fractional landweber method for solving backward time fractional diffusion problem abstract in this paper we propose a fractional landweber regularization method for solving a backward time fractional diffusion problem choice rules for both a priori and a posterior regularization parameters are discussed in deriving the corresponding convergence proofs for the backward problem for verification on the theoretical estimates several numerical examples are constructed and compared with some existing works

recovering the historical distribution for nonlinear space fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space abstract in this paper we investigate the problem of recovering the historical distribution for a nonlinear space fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space this problem is obtained from the classical diffusion equation by replacing the second order space derivative with a fractional laplacian of order 1 2 1 which is usually used to model the anomalous diffusion the problem is severely ill posed to regularize the problem we propose a modified version of the tikhonov regularization method a stability estimate of holder type is established finally several numerical examples based on the finite difference approximation and the discrete fourier transform are presented to illustrate the theoretical results

quasilinear evolutionary equations and continuous interpolation spaces in this paper we analyze the abstract parabolic evolutionary equations dt u x a u u f u h t u 0 x in continuous interpolation spaces allowing a singularity as t 0 here dt denotes the time derivative of order 0 2 we first give a treatment of fractional derivatives in the spaces lp 0 t x and then consider these derivatives in spaces of continuous functions having at most a prescribed singularity as t 0 the corresponding trace spaces are characterized and the dependence on is demonstrated via maximal regularity results on the linear equation dt u x au f u 0 x we arrive at results on existence uniqueness and continuation on the quasilinear equation finally an example is presented

on a quasi reversibility regularization method for a cauchy problem of the helmholtz equation in this paper we consider the cauchy problem for the helmholtz equation in a rectangle where the cauchy data is given for y 0 and boundary data are for x 0 and x p the solution is sought in the interval 0

a fractional tikhonov regularization method for identifying a space dependent source in the time fractional diffusion equation abstract in this paper we are concerned with an inverse source problem for the time fractional diffusion equation with variable coefficients in a general bounded domain the problem is mildly ill posed a new fractional tikhonov method is proposed we discuss the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule and prove the corresponding convergence estimates numerical experiments are conducted for illustrating effectiveness of the proposed method

a note on a cauchy problem for the laplace equation regularization and error estimates abstract in this paper a cauchy problem for the laplace equation is investigated based on the fundamental solution to the elliptic equation we propose to solve this problem by the truncation method which generates well posed problem then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved error estimates for this method are provided together with a selection rule for the regularization parameter the numerical results show that our proposed numerical methods work effectively this work extends to earlier results in qian et al 2008 14 and hao et al 2009 5

determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time abstract we consider the inverse problem of finding the temperature distribution and the heat source whenever the temperatures at the initial time and the final time are given the problem considered is one dimensional and the unknown heat source is supposed to be space dependent only the existence and uniqueness results are proved

some regularization methods for a class of nonlinear fractional evolution equations abstract in this paper we study initial value problems for nonlinear fractional elliptic equations in general the problem is not well posed and herein the hadamard instability occurs under some weak a priori assumptions on the sought solution we propose two new regularization methods to stabilize the problem when the source term is a globally or locally lipschitz function furthermore we also investigate the error estimate and show that the approximate solution converges to the exact solution in l 2 and h 2 norms

on the construction of non invertible minimal skew products abstract let x z be infinite compact metric spaces we show that if the group h z of the homeomorphisms of z has an arc wise connected subgroup h 0 min z whose action on z is minimal then every minimal map f on x invertible or not admits a minimal skew product extension f f g x on x z with the fibre maps g x h 0 min z in the invertible case this was proved by glasner and weiss in 1979 we also contribute to the description of the class z of those spaces z which admit a group h 0 min z with the mentioned property namely we show that this class is closed with respect to countable products and contains all countably infinite products of compact connected manifolds infinitely many of which have nonempty boundary further we show that the subclass of z formed by all compact metric spaces z which admit an arc wise connected group j 0 min z of isometries with a minimal action on z coincides with the class of all homogeneous spaces of compact connected metrizable groups

continuous time random walks and the fractional diffusion equation the average probability density p r t of random walks on fractals is revisited within the continuous time random walks formalism corrections to the accepted asymptotic stretched gaussian decay of p r t of the form ralpha are discussed it is shown that p r t obeys a diffusion equation with a fractional time derivative asymptotically and predictions about the value of alpha are presented

recovering the initial wave amplitude for nonlinear elliptic equation with locally lipschitz source in multiple dimensional domain abstract the backward problems for the elliptic equation bpee are widely used for modelling problems in many fields such as physics geometry or engineering this paper is the first investigation of bpee in unbounded multiple dimensional domain associated with locally lipschitz source as follows 2 u x 1 2 2 u x 2 2 2 u x n 2 2 u y 2 s x 1 x 2 x n y u x 1 x 2 x n y in which x 1 x 2 x n y r n 0 l where l 0 based on the idea of the truncation approach we construct the regularized solution and obtain the h older type of its convergence rate under some assumptions on regularity of the exact solution eventually a numerical experiment for the elliptic allen cahn equation is proposed to illustrate the effectiveness and applicability of our method

on a time space fractional backward diffusion problem with inexact orders abstract in this paper we focus on the backward diffusion problem with the caputo fractional derivative operator in time and a general spatial nonlocal operator for t 0 and s 0 t we consider the problem p s of recovering the distribution u x s from a measure of the final data u x t for the following non homogeneous time space fractional diffusion equation d t u x t k l u x t f x t in r n 0 t subject to the final condition u x t u t x in r n the derivative orders and the nonlocal operator are perturbed with noises firstly for 0 s t we prove the well posedness of problem p s by studying the unique existence and continuity with respect to the derivative orders the source term as well as the final value of the solution secondly for s 0 we verify the ill posedness of problem p 0 and use the method of modified iterated lavrentiev to construct a regularization solution from inexact data and inexact derivative orders we apply a modified form of the discrepancy principle to choose regularization parameter and establish new optimal convergence estimates between the exact solution and its regularized approximation

a particular solution trefftz method for non linear poisson problems in heat and mass transfer this paper proposes a computational procedure based on the trefftz method for the solution of non linear poisson problems the problem is solved by finding an approximate particular solution to the poisson equation and using boundary collocation to solve the resulting laplace equation this method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions the solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency a quasi newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme the efficacy of the new formulation has been demonstrated for two classes of problems viz the thermal explosion problem and the diffusion reaction problem in a partially wetted catalyst pellet a brief error analysis of the method as well as future research directions is presented

initial value boundary value problems for fractional diffusion wave equations and applications to some inverse problems abstract we consider initial value boundary value problems for fractional diffusion wave equation t u x t l u x t where 0 2 where l is a symmetric uniformly elliptic operator with t independent smooth coefficients first we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to and the proofs are based on the eigenfunction expansions second for 0 1 we apply the eigenfunction expansions and prove i stability in the backward problem in time ii the uniqueness in determining an initial value and iii the uniqueness of solution by the decay rate as t iv stability in an inverse source problem of determining t dependent factor in the source by observation at one point over 0 t

ultra analytic effect of cauchy problem for a class of kinetic equations abstract the smoothing effect of the cauchy problem for a class of kinetic equations is studied we firstly consider the spatially homogeneous nonlinear landau equation with maxwellian molecules and inhomogeneous linear fokker planck equation to show the ultra analytic effects of the cauchy problem those smoothing effect results are optimal and similar to heat equation in the second part we study a model of spatially inhomogeneous linear landau equation with maxwellian molecules and show the analytic effect of the cauchy problem

the boundary element method for the solution of the backward heat conduction equation in this paper we consider the numerical solution of the one dimensional unsteady heat conduction equation in which dirichlet boundary conditions are specified at two space locations and the temperature distribution at a particular time say t0 is given the temperature distribution for all times t t0 is now required and this backward heat conduction problem is a well known improperly posed problem in order to solve this problem the minimal energy technique has been introduced in order to modify the boundary element method and this results in a stable approximation to the solution and the accuracy of the numerical results are very encouraging

a discontinuous petrov galerkin method for time fractional diffusion equations we propose and analyze a time stepping discontinuous petrov galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time fractional subdiffusion problems we prove the existence uniqueness and stability of approximate solutions and derive error estimates to achieve high order convergence rates from the time discretizations the time mesh is graded appropriately near t 0 to compensate for the singular temporal behavior of the exact solution near t 0 caused by the weakly singular kernel but the spatial mesh is quasi uniform in the l infty 0 t l 2 omega norm 0 t is the time domain and omega is the spatial domain for sufficiently graded time meshes a global convergence of order k m alpha 2 h r 1 is shown where 0 alpha 1 is the fractional exponent k is the maximum time step h is the maximum diameter of the elements of the spatial mesh and m and r are the degrees of approximate solutions in time an

multiscaling fractional advection dispersion equations and their solutions 1 the multiscaling fractional advection dispersion equation ade is a multidimensional model of solute transport that encompasses linear advection fickian dispersion and super fickian dispersion the super fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions the directions need not be orthogonal so the model can be applied to irregular noncontinuum fracture networks the statistical model underlying multiscaling fractional dispersion is a continuous time random walk ctrw in which particles have arbitrary jump length distributions and finite mean waiting time distributions the meaning of the parameters in a compound poisson process a subset of ctrws is used to develop a physical interpretation of the equation variables the green s function solutions are the densities of operator stable probability distributions the limit distributions of normalized sums of independent and identically distributed random vectors these densities can be skewed heavy tailed and scale nonlinearly resembling solute plumes in granular aquifers they can also have fingers in any direction resembling transport along discrete pathways such as fractures

topological conjugacy and transitivity for a class of piecewise monotone maps of the interval on dit qu une application continue f d un intervalle compact dans lui meme est markov lineaire s il est lineaire par morceaux et l ensemble de tous les f k x ou k 0 et x une extremite d un morceau lineaire est fini on donne une classification effective pour des applications de markov lineaires et une procedure effective pour determiner quand une telle application est transitive

two numerical methods for solving a backward heat conduction problem we introduce a central difference method and a quasi reversibility method for solving a backward heat conduction problem bhcp numerically for these two numerical methods we give the stability analysis meanwhile we investigate the roles of regularization parameters in these two methods numerical results show that our algorithm is effective

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

osculatory interpolation in the method of fundamental solution for nonlinear poisson problems abstract the method of fundamental solution also known as the f trefftz method or the singularity method is an efficient numerical method for the solution of laplace equation for both two and three dimensional problems in recent years the method has also been applied for the solution of poisson equations by finding the particular solution to the nonhomogeneous terms in general approximate particular solutions are constructed using the interpolation of the nonhomogeneous terms by the radial basis functions the method has been validated in recent papers this paper presents an improvement of the solution procedure for such problems the improvement is achieved by using radial basis functions called osculatory radial basis functions such functions make use of the normal gradient at boundary to obtain improved interpolation the efficacy of the method is demonstrated for some prototypical nonlinear poisson problems and for multiple poisson equations

chaos periodicity and snakelike continua the results of this paper relate the dynamics of a continuous map f of the interval and the topology of the inverse limit space with bonding map f these inverse limit spaces have been studied by many authors and are examples of what bing has called snakelike continua roughly speaking we show that when the dynamics of f are complicated the inverse limit space contains indecomposable subcontinua we also establish a partial converse

the method of simplified tikhonov regularization for dealing with the inverse time dependent heat source problem this paper investigates the inverse problem of determining a heat source using a parabolic equation where data are given at some fixed location the problem is ill posed i e the solution if it exists does not depend continuously on the data a simplified tikhonov regularization method is given and an order optimal stability estimate is obtained a numerical example shows that the regularization method is effective and stable

cauchy problem for fractional diffusion equations abstract we consider an evolution equation with the regularized fractional derivative of an order 0 1 with respect to the time variable and a uniformly elliptic operator with variable coefficients acting in the spatial variables such equations describe diffusion on inhomogeneous fractals a fundamental solution of the cauchy problem is constructed and investigated

some uniqueness and existence results for the initial boundary value problems for the generalized time fractional diffusion equation in this paper some uniqueness and existence results for the solutions of the initial boundary value problems for the generalized time fractional diffusion equation over an open bounded domain gx 0 t email protected r n are given to establish the uniqueness of the solution a maximum principle for the generalized time fractional diffusion equation is used in turn the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is considered in the paper too another important consequence of the maximum principle is the continuous dependence of the solution on the problem data to show the existence of the solution the fourier method of the variable separation is used to construct a formal solution under certain conditions the formal solution is shown to be a generalized solution of the initial boundary value problem for the generalized time fractional diffusion equation that turns out to be a classical solution under some additional conditions

regularization of cauchy abstract problem for a coupled system for nonlinear elliptic equations abstract in this paper we consider a cauchy problem of coupled elliptic equations we show that the problem is ill posed in the sense of hadamard and a regularization is in order under some weak a priori assumptions on the exact solution we propose a new regularization method to stabilize the problem when the source term is a global or local lipschitz function furthermore we also obtain error estimates between the regularized solution and the sought solution

filter regularization for final value fractional diffusion problem with deterministic and random noise abstract in this paper we consider an inverse problem for a time fractional diffusion equation with inhomogeneous source to determine the initial data from the observation data provided at a later time in general this problem is ill posed therefore we construct a regularized solution using the filter regularization method in both cases the deterministic case and random noise case first we propose both parameter choice rule methods the a priori and the a posteriori methods then we obtain the convergence rates and provide examples of filters we also provide a numerical example to illustrate our results

numerical solution of forward and backward problem for 2 d heat conduction equation for a two dimensional heat conduction problem we consider its initial boundary value problem and the related inverse problem of determining the initial temperature distribution from transient temperature measurements the conditional stability for this inverse problem and the error analysis for the tikhonov regularization are presented an implicit inversion method which is based on the regularization technique and the successive over relaxation sor iteration process is established due to the explicit difference scheme for a direct heat problem developed in this paper the inversion process is very efficient while the application of sor technique makes our inversion convergent rapidly numerical results illustrating our method are also given

simplified tikhonov and fourier regularization methods on a general sideways parabolic equation the inverse heat conduction problem ihcp can be considered to be a sideways parabolic equation in the quarter plane and now the results available in the literature on ihcp mainly devoted to the standard sideways heat equation numerical methods have been developed also for more general equations but in most cases the stability theory and convergence proofs have not been generalized accordingly this paper remedies this by a simplified tikhonov and a new fourier regularization methods on a general sideways parabolic equation some known results for sideways heat equation are only the special case of the conclusions in this paper the numerical example shows that the computation effect is satisfactory

rapid fluctuations of chaotic maps on rn the iterates f n of a chaotic map f display heightened oscillations or fluctuations as n i ff is a chaotic interval map in one dimension then it is now known that the total variation of f n on that interval grows exponentially with respect to n g chen t huang y huang chaotic behavior of interval maps and total variations of iterates internat j bifur chaos 14 2004 2161 2186 however the characterization of chaotic behavior of maps in multi dimensional spaces is generally much more challenging here we generalize the definition of bounded variations for vector valued maps in terms of the hausdorff measure and then use it to study what we call rapid fluctuations on fractal sets in multi dimensional chaotic discrete dynamical systems the relations among rapid fluctuations strict turbulence and positive entropy are established for lipschitz continuous systems on general n dimensional euclidean spaces applications to planar monotone or competitive systems and triangular systems on the square are also given

a new general filter regularization method for cauchy problems for elliptic equations with a locally lipschitz nonlinear source up to now studies on the semi linear cauchy problem for elliptic partial differential equations needed to assume that the source term present in the governing equation is a global lipschitz function the current paper is the first investigation to not only the more general but also the more practical case of interest when the source term is only a local lipschitz function in such a situation the methods of solution from the previous studies with a global lipschitz source term are not directly applicable and therefore novel ideas and techniques need to be developed to tackle the local lipschitz nonlinearity this locally lipschitz source arises in many applications of great physical interest governed by for example the sine gordon lane emden allen cahn and liouville equations the inverse problem is severely ill posed in the sense of hadamard by violating the continuous dependence upon the input cauchy data therefore in order to obtain a stable solution we consider theoretical aspects of regularization of the problem by a new generalized filter method under some priori assumptions on the exact solution we prove and obtain rigorously convergence estimates

fourth order modified method for the cauchy problem for the laplace equation we consider the cauchy problem for the laplace equation in the half plane x 0 y r where the cauchy data is given at x 0 and the solution is sought in the interval 0 x 1 the problem is ill posed the solution if it exists does not depend continuously on the data in order to solve the problem numerically it is necessary to modify the equation so that a bound on the solution is imposed we study a modification of the equation where a fourth order mixed derivative term is added error estimates for this equation are given which show that the solution of the modified equation is an approximation of the solution of the cauchy problem for the laplace equation and it is shown that when the data error tends to zero the error in the approximate solution tends to zero logarithmically numerical implementation is considered and a simple example is given

on nonlocal backward problems for fractional stochastic diffusion equations abstract this paper is devoted to solving a nonlocal backward problem for fractional stochastic diffusion equations based on the eigenfunction expansion of the solution the nonlocal backward problem for searching the previous values is reduced to an integral equation we obtain the boundedness strong continuity and compactness of the new operators generated by the nonlocally terminal value term and then we proceed to obtain the existence of mild solutions for the nonlocal backward problem

the nilpotency class of p groups in which subgroups have few conjugates abstract we consider the problem of how the nilpotency class of a finite p group can be bounded in terms of the maximum length of the conjugacy classes of cyclic subgroups we sharpen some previously known bounds and also prove that a p group in which every cyclic subgroup has at most p 2 conjugates has class at most 4

on a fractional backward heat conduction problem application to deblurring in this paper we consider a fractional backward heat conduction problem fbhcp in the two dimensional space which is associated with a deblurring problem from three different perspectives we give all kinds of regularization methods based on a general form of regularization methods we devise some new methods deblurring experiment shows that the new regularization methods work well for restoration of images

sensitive dependence for nonautonomous discrete dynamical systems abstract given a nonautonomous discrete dynamical system nds x f 1 we show that transitivity and density of periodic points do not imply sensitivity in general i e in the definition of devaney chaos there are no redundant conditions for nds in addition we show that if we also assume uniform convergence of the sequence f n that induces the nds then sensitivity follows furthermore in contrast to the autonomous case we show that there exist minimal nds which are neither equicontinuous nor sensitive

on a backward problem for inhomogeneous time fractional diffusion equations abstract in this paper we consider a final value problem for time fractional diffusion equation with inhomogeneous source the main goal of our paper is to determine an approximated initial data from the observation data at final time by constructing a regularized solution using a mollification method under appropriate regularity assumptions of the exact solution we give convergence rate between the reconstructed solution and the exact one we also provide a numerical example to illustrate the main results

a computational quasi reversiblility method for cauchy problems for laplace s equation this work concerns the use of the method of quasi reversibility for solving cauchy problems for laplace s equation the paper begins with a derivation of an error estimate for this method using car

a mollification method for ill posed problems a mollification method for a class of ill posed problems is suggested the idea of the method is very simple and natural if the data are given inexactly then we try to find a sequence of mollification operators which map the improper data into well posedness classes of the problem mollify the improper data within these mollified data our problem becomes well posed and when these facts are in hand we try to obtain error estimates and optimal or quasi optimal mollification parameters the method is working not only for problems in hilbert spaces but also for problems in banach spaces applications of the method to concrete problems like numerical differentiation parabolic equations backwards in time the cauchy problem for the laplace equation one and multidimensional non characteristic cauchy problems for parabolic equations in infinite or finite domains give us very sharp stability estimates of h older continuous type in these cases the method is optimal in the sense that it gives the same order of h older continuous dependence on the data as for the regularized problems furthermore the method may be implemented numerically using fast fourier transforms for the first time a uniform stability estimate of h older continuous type of the solution of the heat equation backwards in time in the space l p bbb r for all p in 1 infty could be established by our mollification method a new simple sharp pointwise estimate of h older type for the weak solution of a non characteristic cauchy problem for parabolic equations in a finite domain is established

p groups with maximal elementary abelian subgroups of rank 2 abstract let p be an odd prime number and g a finite p group we prove that if the rank of g is greater than p then g has no maximal elementary abelian subgroup of rank 2 it follows that if g has rank greater than p then the poset e g of elementary abelian subgroups of g of rank at least 2 is connected and the torsion free rank of the group of endotrivial kg modules is one for any field k of characteristic p we also verify the class breadth conjecture for the p groups g whose poset e g has more than one component

a modified method for a backward heat conduction problem we consider a backward heat conduction problem in a strip where data is given at the final time t t t 0 and a solution for 0 t t is sought the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data in order to numerically solve the problem we study a modification of the equation where a third order mixed derivative term is added error estimates for this problem are given which show that the modified problem is stable and its solution is an approximation of the backward heat conduction problem some numerical tests illustrate that the proposed method is feasible and effective c 2006 elsevier inc all rights reserved

a kernel based method for the approximate solution of backward parabolic problems a method for approximating the solution of self adjoint parabolic equations backwards in time is presented the technique uses the green s function of the parabolic equation to formulate the problem as a first kind fredholm integral equation and then does a regularized least squares solution of the resulting discrete system error estimates and examples are presented direct knowledge of the kernel of the differential operator is not required

graded lie algebras of maximal class we study graded lie algebras of maximal class over a field mathbf f of positive characteristic p a shalev has constructed infinitely many pairwise non isomorphic insoluble algebras of this kind thus showing that these algebras are more complicated than might be suggested by considering only associated lie algebras of p groups of maximal class here we construct mathbf f aleph 0 pairwise non isomorphic such algebras and max mathbf f aleph 0 soluble ones both numbers are shown to be best possible we also exhibit classes of examples with a non periodic structure as in the case of groups two step centralizers play an important role

maximum principle for the generalized time fractional diffusion equation abstract in the paper a maximum principle for the generalized time fractional diffusion equation over an open bounded domain g 0 t g r n is formulated and proved the proof of the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is given in the paper too the maximum principle is then applied to show that the initial boundary value problem for the generalized time fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions

a non local boundary value problem method for parabolic equations backward in time abstract the ill posed parabolic equation backward in time u t a u 0 0 t t u t f subject to the constraint u 0 e with the positive self adjoint unbounded operator a and e 0 being given is regularized by the well posed non local boundary value problem u t a u 0 0 t t u 0 u t f 0 the error estimates of holder type of the regularized solutions are obtained these estimates improve the related results by mel nikova denche and bessila

optimal error bound and simplified tikhonov regularization method for a backward problem for the time fractional diffusion equation in this paper we consider a backward problem for a time fractional diffusion equation such a problem is ill posed the optimal error bound for the problem under a source condition is analyzed a simplified tikhonov regularization method is utilized to solve the problem and its convergence rates are analyzed under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule respectively numerical examples show that the proposed regularization method is effective and stable and both parameter choice rules work well

a splitting theorem for transitive maps in this paper we prove a splitting theorem for continuous transitive maps on locally connected compact metric spaces which generalizes the results of barge and martin 4 for transitive maps on the interval and of blokh 7 for transitive graph maps as a consequence lower bounds for the topological entropy of transitive graph maps in terms of the cardinality of their splittings are obtained also we show that for every connected graph which is not a tree the infimum of the topological entropy of the transitive maps having a periodic point is zero since the topological entropy of transitive maps without periodic points is zero 7 and positive lower bounds of the entropy were given for the transitive maps on trees 2 this paper completes the problem of obtaining lower bounds of the topological entropy for transitive maps of any connected graph

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

optimal filtering for the backward heat equation for the backward heat equation stabilized by an a priori initial bound an estimator is determined for intermediate values that is optimal with respect to the bound and the observation accuracy it is shown how this may be implemented computationally with error estimates for the computed approximation which can be made arbitrarily close to the uncertainty level induced by the ill posedness of the underlying problem thus the feasibility of this for practical computation inevitably severely limited by that inherent uncertainty is as good as possible

on tikhonov s method and optimal error bound for inverse source problem for a time fractional diffusion equation abstract we investigate the linear but ill posed inverse problem of determining a multi dimensional space dependent heat source in a time fractional diffusion equation we show that the problem is ill posed in the hilbert scale h r r n and establish global order optimal lower bound for the worst case error next we use the tikhonov regularization method to deal with this problem in the hilbert scale h r r n locally optimal choices of parameters for the family of regularization operator in the hilbert scales h r r n are analyzed by a priori and a posteriori methods numerical implementations are presented to illustrate our theoretical findings

topological entropy of fuzzified dynamical systems a discrete dynamical system is given by a compact metric space x and any continuous self map defined on x this discrete dynamical system can be naturally extended to the space of fuzzy sets on x in this paper we study relations between the sizes of the topological entropies of the original dynamical system and of its fuzzy counterpart among other things we present a constructive proof of the fact that even very weak assumptions on the crisp discrete dynamical system ensure infinite topological entropy of the fuzzy system however we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system

on a riesz feller space fractional backward diffusion problem with a nonlinear source abstract in this paper a backward diffusion problem for a space fractional diffusion equation with a nonlinear source in a strip is investigated this problem is obtained from the classical diffusion equation by replacing the second order space derivative with a riesz feller derivative of order 0 2 a nonlinear problem is severely ill posed therefore we propose two new modified regularization solutions to solve it we further show that the approximated problems are well posed and their solutions converge if the original problem has a classical solution in addition the convergence estimates are presented under a priori bounded assumption of the exact solution for estimating the error of the proposed method a numerical example has been implemented