generalized aor method for solving a class of generalized saddle point problems abstract in this paper by constructing appropriate splitting a generalized aor gaor method for solving a class of generalized saddle point problems has been presented and the convergence of the method is studied the numerical experiments results show the efficiency and robustness of the new method

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

spatially periodic stochastic system with infinite globally coupled oscillators in this paper we study a spatially periodic stochastic system with infinite globally coupled oscillators driven by a constant force f with two typical models we show that when f 0 there is a nonequilibrium transition between the state with zero mean field s 0 and the state with nonzero mean field s not equal 0 for model i the transition is not a phase transition while for the model ii it is second order in addition we find that for coupled oscillators driven only by additive noises when f 0 a transport may emerge if the nonzero mean field breaks the symmetry of the systems with varying f a continuous or discontinuous transition between state s 0 and state s 0 will appear the mean field or current sometimes exhibits hysteresis as a function of f

dynamic mean field models h theorem for stochastic processes and basins of attraction of stationary processes abstract stochastic processes of dynamic mean field models are studied in terms of nonlinear fokker planck equations we show that h theorems for single time point distributions can be used to derive h theorems for hierarchies of joint distributions in doing so we prove the convergence of transient stochastic processes to stationary ones for multistable systems we furthermore determine the basins of attraction of these stationary processes our results are illustrated by means of a mean field model exhibiting bistability

block triangular preconditioners for stabilized saddle point problems with nonsymmetric 1 1 block abstract we study the performance of a class of block triangular preconditioners for saddle point systems with nonsymmetric positive definite 1 1 block the presented results incorporate the established results in zhang and zhao 2018 where a parameter dependent preconditioner was suggested to solve the mentioned saddle point systems the performance of the preconditioner in conjunction with krylov subspace methods is examined for solving a test problem arising from stabilized finite element discretizations of the oseen problem the reported numerical experiments show that there exist parameter free preconditioners which outperform the preconditioner proposed in the earlier referred work

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

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

results on residual renyi entropy of order statistics and record values this paper explores properties of the residual renyi entropy of some ordered random variables the residual renyi entropy of the kth order statistic from a continuous distribution function is represented in terms of the residual renyi entropy of the kth order statistic from uniform distribution the monotone behavior of the residual renyi entropy of order statistic under various conditions is discussed analogues results for the residual renyi entropy of record values are also given

on inexact preconditioners for nonsymmetric matrices inexact versions of the block triangular preconditioners for nonsymmetric matrices of block two by two structures in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 and i c f ipsen siam j sci comput 23 2001 pp 1050 1051 are presented and the two preconditioners for symmetric block two by two matrices in c durazzi and v ruggiero numer linear algebra appl 10 2003 pp 673 688 are extended to general nonsymmetric matrices moreover we precisely describe the spectral properties of the preconditioned matrices and the finite step termination properties of the preconditioned krylov subspace iteration methods with an optimal or galerkin property with respect to these preconditioners several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners

on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems in this paper we first present a local hermitian and skew hermitian splitting lhss iteration method for solving a class of generalized saddle point problems the new method converges to the solution under suitable restrictions on the preconditioning matrix then we give a modified lhss mlhss iteration method and further extend it to the generalized saddle point problems obtaining the so called generalized mlhss gmlhss iteration method numerical experiments for a model navier stokes problem are given and the results show that the new methods outperform the classical uzawa method and the inexact parameterized uzawa method

a relaxed dimensional factorization preconditioner for the incompressible navier stokes equations in this paper we introduce a relaxed dimensional factorization rdf preconditioner for saddle point problems properties of the preconditioned matrix are analyzed and compared with those of the closely related dimensional splitting ds preconditioner recently introduced by benzi and guo 7 numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the rdf preconditioner with respect to both mesh size and viscosity

on the solutions of nonlinear fractional klein gordon equation with modified riemann liouville derivative abstract in this paper fractional variational iteration method fvim is applied for finding approximate analytical solutions of nonlinear fractional klein gordon equation time fractional derivative is described in the riemann liouville sense a new application of fractional variational iteration method fvim is extended to derive analytical solutions in the form of a series for this equation the behavior of the solutions and the effects of different values of fractional order are indicated graphically it is shown that the solutions obtained by the fvim are reliable convenient and effective for strongly nonlinear partial equations with modified riemann liouville derivative

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

a wavelet operational method for solving fractional partial differential equations numerically fractional calculus is an extension of derivatives and integrals to non integer orders and a partial differential equation involving the fractional calculus operators is called the fractional pde they have many applications in science and engineering however not only the analytical solution existed for a limited number of cases but also the numerical methods are very complicated and difficult in this paper we newly establish the simulation method based on the operational matrices of the orthogonal functions we formulate the operational matrix of integration in a unified framework by using the operational matrix of integration we propose a new numerical method for linear fractional partial differential equation solving in the method we 1 use the haar wavelet 2 establish a lyapunov type matrix equation and 3 obtain the algebraic equations suitable for computer programming two examples are given to demonstrate the simplicity clarity and powerfulness of the new method

the polya information divergence extensions of sanov s theorem and the conditional limit theorem colt are established for a multicolor polya eggenberger pe urn sampling scheme giving the polya information divergence and a polya extension to the maximum relative entropy maxent method polya maxent includes the standard maxent as well as its variants used in bose einstein fermi dirac and intermediate acharya swamy statistics as special cases in the pe setting standard maxent is in general asymptotically inconsistent

eigenvalue estimates for saddle point matrices of hermitian and indefinite leading blocks we study the eigenvalue bounds for the nonsingular saddle point matrices of hermitian and indefinite 1 1 and 2 2 blocks without imposing the restrictions that the 1 1 blocks are positive definite on the kernels of the 2 1 blocks

lie symmetry analysis conservation laws and numerical approximations of time fractional fokker planck equations for special stochastic process in foreign exchange markets abstract in this paper the transition joint probability density function of the solution of the ornstein uhlenbeck process is presented by a deterministic parabolic time fractional pde fpde named time fractional fokker planck equation the article generally is divided to two parts theoretical and approximated analysis in the theoretical sections lie group method and invariant subspace method are applied for finding exact solutions and conservation laws of the considered equation in the next part by chebyshev wavelets s method the numerical solutions are driven then the usefulness of this approximated method is comparing with the exact solutions by some plotted graphs

fractional variational calculus for nondifferentiable functions we prove necessary optimality conditions in the class of continuous functions for variational problems defined with jumarie s modified riemann liouville derivative the fractional basic problem of the calculus of variations with free boundary conditions is considered as well as problems with isoperimetric and holonomic constraints

on the fractional solution of the equation f x y f x f y and its application to fractional laplace s transform abstract it is shown that if the problem is defined in the setting of fractional calculus via fractional difference on non differentiable functions then the solution of the functional equation f x y f x f y is exactly defined as the solution of a linear fractional differential equation the dual or counterpart problem that is the fractional solution of the equation g xy g x g y is also considered and it is shown that the corresponding solution is the logarithm of fractional order defined as the inverse of a generalized mittag leffler function which is nowhere differentiable this framework suggests a definition of fractional laplace s transform expressed in terms of generalized mittag leffler function and its main properties are outlined mainly inverse function and convolution one takes this opportunity to display a new fractional taylor s series for functions f x y of two variables x and y many open problems are stated which are directly related to the non differentiability of the functions so involved therefore the title on the fractional solution

a note on the positive stable block triangular preconditioner for generalized saddle point problems abstract in this paper we give a sharp eigenvalue bound for the positive stable block triangular preconditioned matrix presented in a recent paper by cao z h cao positive stable block triangular preconditioners for symmetric saddle point problems appl numer math 57 2007 899 910 the intervals containing these eigenvalues of the preconditioned matrix remove the origin which is benefit for further study numerical experiments of a model stokes problem are presented to show the estimate and the effectiveness of the positive stable block triangular preconditioners

derivation of a new mertons optimal problem presented by fractional stochastic stock price and its applications in this article a new model of mertons optimal problem is derived this derivation is based on stock price presented by fractional order stochastic differential equation an extension of hamiltonjacobibellman is used to transfer our proposed model to a fractional partial differential equation as an application of our proposed model two optimal problems are discussed and solved analytically

stochastic bifurcation in fitzhugh nagumo ensembles subjected to additive and or multiplicative noises abstract we have studied the dynamical properties of finite n unit fitzhugh nagumo fn ensembles subjected to additive and or multiplicative noises reformulating the augmented moment method amm with the fokker planck equation fpe method h hasegawa j phys soc japan 75 2006 033001 in the amm original 2 n dimensional stochastic equations are transformed to eight dimensional deterministic ones and the dynamics is described in terms of averages and fluctuations of local and global variables the stochastic bifurcation is discussed by a linear stability analysis of the deterministic amm equations the bifurcation transition diagram of multiplicative noise is rather different from that of additive noise the former has the wider oscillating region than the latter the synchronization in globally coupled fn ensembles is also investigated results of the amm are in good agreement with those of direct simulations dss

a nonlinear fractional sharma tasso olver equation due to its importance in scientific applications the fractional nonlinear sharma tasso olver equation has been studied by several authors from point of view of exact solutions in this paper new explicit and exact solutions are obtained by using the fractional complex transform and the improved extended tanh coth method the results can be compared with other methods used to solve the same model

preconditioning for the steady state navier stokes equations with low viscosity we introduce a preconditioner for the linearized navier stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero for constant coefficient problems with periodic boundary conditions we show that the preconditioning yields a system with a single eigenvalue equal to 1 so that performance is independent of both viscosity and mesh size for other boundary conditions we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon we also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration with virtually no degradation of performance

new local generalized shift splitting preconditioners for saddle point problems based on a local generalized shift splitting of the saddle point matrix with symmetric positive definite 1 1 block and symmetric positive semidefinite 2 2 block a new local generalized shift splitting preconditioner with two shift parameters for solving saddle point problems is proposed the preconditioner is extracted from a new local generalized shift splitting iteration and can lead to the unconditional convergence of the iteration in addition we consider solving the saddle point systems by preconditioned krylov subspace methods and discuss some properties of the preconditioned saddle point matrix with a deteriorated preconditioner such as eigenvalues eigenvectors and degree of the minimal polynomial numerical experiments arising from a finite element discretization model of the stokes problem are given to validate the effectiveness of the proposed preconditioner

lie symmetry analysis explicit solutions and conservation laws for the space time fractional nonlinear evolution equations abstract this paper studies the symmetry analysis explicit solutions convergence analysis and conservation laws cls for two different space time fractional nonlinear evolution equations with riemann liouville rl derivative the governing equations are reduced to nonlinear ordinary differential equation ode of fractional order using their lie point symmetries in the reduced equations the derivative is in erdelyi kober ek sense power series technique is applied to derive an explicit solutions for the reduced fractional odes the convergence of the obtained power series solutions is also presented moreover the new conservation theorem and the generalization of the noether operators are developed to construct the nonlocal cls for the equations some interesting figures for the obtained explicit solutions are presented

stability analysis of stationary states of mean field models described by fokker planck equations abstract so far the stability of stationary solutions of mean field models described by fokker planck equations has only been determined analytically in some special cases following two earlier studies j korean phys soc 40 2002 1037 prog theoret phys suppl 150 2003 48 we discuss a stability analysis for a large class of mean field models based on fokker planck equations to this end we use linear stability analysis in addition to the well known approaches by means of transcendent equations and lyapunov s direct method we demonstrate that all three methods yield consistent results for systems that exhibit free energy functionals e g equilibrium systems we show that the simple transcendent equation analysis fails for systems that do not exhibit free energy functionals e g nonequilibrium systems and show how to solve this problem by means of a more sophisticated transcendent equation analysis furthermore we propose a norm for the perturbations of stationary states and illustrate some of our results by a model that exhibits a reentrant noise induced phase transition

generalized einstein relation and the metzler klafter conjecture in a composite subdiffusive regime abstract in this paper a generalized diffusion model driven by the composite subdiffusive fractional brownian motion fbm is employed based on this stochastic process we derive a fractional fokker planck equation ffpe and obtain its solution it is proved that the generalized einstein relation ger and the metzler and klafter conjecture on the asymptotic behavior of stretched gaussian hold the ffpe in a composite subdiffusive regime

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

a modified dimensional split preconditioner for generalized saddle point problems abstract for large and sparse saddle point linear systems arising from 2d linearized navier stokes equations benzi and guo recently studied a dimensional split ds preconditioner appl numer math 61 2011 66 76 by further applying it to generalized saddle point problems in this paper we present a modified dimensional split mds preconditioner this new preconditioner is based on a splitting of the generalized saddle point matrix resulting in an unconditional convergent fixed point iteration the basic iteration is accelerated by a krylov subspace method like restarted gmres the implementation of the mds preconditioner is discussed and a similar case is also analyzed finally numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the mds preconditioner

uncertainly measures of rough set prediction abstract the main statistics used in rough set data analysis the approximation quality is of limited value when there is a choice of competing models for predicting a decision variable in keeping within the rough set philosophy of non invasive data analysis we present three model selection criteria using information theoretic entropy in the spirit of the minimum description length principle our main procedure is based on the principle of indifference combined with the maximum entropy principle thus keeping external model assumptions to a minimum the applicability of the proposed method is demonstrated by a comparison of its error rates with results of c4 5 using 14 published data sets

analysis of lie symmetries with conservation laws and solutions for the generalized 3 1 dimensional time fractional camassa holm kadomtsev petviashvili equation abstract in this paper under investigated is a generalized 3 1 dimensional camassa holm kadomtsev petviashvili gch kp equation which describes the role of dispersion in the formation of patterns in liquid drops with the help of the semi inverse method the euler lagrange equation and agrawal s method the time fractional gch kp equation is derived in the sense of riemann liouville fractional derivatives further the symmetry of the 3 1 dimensional time fractional gch kp equation is studied by fractional order symmetry meanwhile based on the new conservation theorem the conservation laws of 3 1 dimensional time fractional gch kp equation are constructed finally the solutions to the equation are given via a bilinear method and the radial basis functions rbfs meshless approach

a block positive semidefinite splitting preconditioner for generalized saddle point linear systems abstract in this paper we propose a new block positive semidefinite splitting bps preconditioner for a class of generalized saddle point linear systems the new bps preconditioner is based on two positive semidefinite splittings of the generalized saddle point matrix resulting in an unconditional convergent fixed point iteration method theoretical results show that all eigenvalues of the bps preconditioned matrix are clustered at only two points as the iteration parameter is close to zero two numerical examples arising from the mixed finite element discretization of the linearized navier stokes equation and the meshfree discretization of the piezoelectric structure equation are used to illustrate the effectiveness of the new preconditioner

modified riemann liouville derivative and fractional taylor series of nondifferentiable functions further results the paper gives some results and improves the derivation of the fractional taylor s series of nondifferentiable functions obtained recently in the form f g h e a h ad g a f g 0 a 1 where e a is the mittag leffier function here one defines fractional derivative as the limit of fractional difference and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using riemann liouville definition as a result a modified riemann liouville definition is proposed which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one s in order to support this f taylor series one shows how its first term can be obtained directly in the form of a mean value formula the fractional derivative of the dirac delta function is obtained together with the fractional taylor s series of multivariate functions the relation with irreversibility of time and symmetry breaking is exhibited and to some extent this f taylor s series generalizes the fractional mean value formula obtained a few years ago by kolwantar

improving generalization of fuzzy if then rules by maximizing fuzzy entropy when fuzzy if then rules initially extracted from data have not a satisfying performance we consider that the rules require refinement distinct from most existing rule refinement approaches that are based on the further reduction of training error this paper proposes a new rule refinement scheme that is based on the maximization of fuzzy entropy on the training set the new scheme which is realized by solving a quadratic programming problem is expected to have the advantages of improving the generalization capability of initial fuzzy if then rules and simultaneously overcoming the overfitting of refinement experimental results on a number of selected databases demonstrate the expected improvement of generalization capability and the prevention of overfitting by a comparison of both training and testing accuracy before and after the refinement

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

colored noise induced discontinuous transitions in symbiotic ecosystems a symbiotic ecosystem is studied by means of the lotka volterra stochastic model using the generalized verhulst self regulation the effect of fluctuating environment on the carrying capacity of a population is taken into account as dichotomous noise the study is a follow up of our investigation of symbiotic ecosystems subjected to three level trichotomous noise r mankin a ainsaar a haljas and e reiter phys rev e 65 051108 2002 relying on the mean field theory an exact self consistency equation for stationary states is derived in some cases the mean field exhibits hysteresis as a function of noise parameters it is established that random interactions with the environment can cause discontinuous transitions the dependence of the critical coupling strengths on the noise parameters is found and illustrated by phase diagrams predictions from the mean field theory are compared with the results of numerical simulations our results provide a possible scenario for catastrophic shifts of population sizes observed in nature

a new relaxed pss preconditioner for nonsymmetric saddle point problems a new relaxed pss like iteration scheme for the nonsymmetric saddle point problem is proposed as a stationary iterative method the new variant is proved to converge unconditionally when used for preconditioning the preconditioner differs from the coefficient matrix only in the upper right components the theoretical analysis shows that the preconditioned matrix has a well clustered eigenvalues around 1 0 with a reasonable choice of the relaxation parameter this sound property is desirable in that the related krylov subspace method can converge much faster which is validated by numerical examples

regularized dpss preconditioners for generalized saddle point linear systems abstract by introducing a regularization matrix and an additional iteration parameter a new class of regularized deteriorated positive definite and skew hermitian splitting rdpss preconditioners are proposed for generalized saddle point linear systems compared with the well known hermitian and skew hermitian splitting hss preconditioner and the regularized hss preconditioner bai 2019 studied recently the new rdpss preconditioners have much better computing efficiency especially when the 1 1 block matrix is non hermitian it is proved that the corresponding rdpss stationary iteration method is unconditionally convergent in addition clustering property of the eigenvalues of the rdpss preconditioned matrix is studied in detail two numerical experiments arising from the meshfree discretization of a static piezoelectric equation and the finite element discretization of the navier stokes equation show the effectiveness of the new proposed preconditioners

analytical approximate solutions for nonlinear fractional differential equations we consider a class of nonlinear fractional differential equations fdes based on the caputo fractional derivative and by extending the application of the adomian decomposition method we derive an analytical solution in the form of a series with easily computable terms for linear equations the method gives exact solution and for non linear equations it provides an approximate solution with good accuracy several examples are discussed

a mini tutorial on measure valued markov processes and nonlinear martingale problems as a reply to mccauley s comment abstract one goal of this mini tutorial is to provide an introduction into the theory of measure valued markov processes and nonlinear martingales defined by strongly nonlinear fokker planck equations and to discuss the physical relevance of the associated processes another goal is to reply to mccauley s comment on t d frank physica a 331 391 2004 the tutorial addresses in detail two approaches found in physics and mathematics the first approach exploits a mapping between linear and nonlinear fokker planck equations the second approach exploits martingale theory several examples of markov processes and martingales in quantum mechanical nonextensive and self organizing systems defined by nonlinear fokker planck equations are discussed

a new matrix splitting preconditioner for generalized saddle point problems abstract for generalized saddle point problems we establish a new matrix splitting preconditioner and give the implementing process in detail the new preconditioner is much easier to be implemented than the modified dimensional split mds preconditioner the convergence properties of the new splitting iteration method are analyzed the eigenvalue distribution of the new preconditioned matrix is discussed and an upper bound for the degree of its minimal polynomial is derived finally some numerical examples are carried out to verify the effectiveness and robustness of our preconditioner on generalized saddle point problems discretizing the incompressible navier stokes equations

derivation of an amplitude of information in the setting of a new family of fractional entropies by generalizing the basic functional equation f xy f x f y in the form f b xy f b x f b y b 1 one can derive a family of solutions which are exactly the inverse of the mittag leffler function referred to as mittag leffler logarithm or logarithm of fractional order this result provides a new family of generalized informational entropies which are indexed by a parameter clearly related to fractals via fractional calculus and which is quite relevant in the presence in defect of observation the relation with shannon s entropy renyi s entropy and tsallis entropy is clarified and it is shown that tsallis generalized logarithm has a significance in terms of fractional calculus the case b 2 looks like directly relevant to amplitude of probability in quantum mechanics and provides an approach to the definition of amplitude of informational entropy one examines the kind of result one can so obtain in applying the maximum entropy principle in the presence of uncertain definition or fuzzy definition the mittag leffler function would be more relevant than the gaussian normal law to some extent this new formulation could be fully supported by the derivation of a new family of fractional fisher information

two modified block triangular splitting preconditioners for generalized saddle point problems abstract for the generalized saddle point problems firstly we introduce a modified generalized relaxed splitting mgrs preconditioner to accelerate the convergence rate of the krylov subspace methods based on a block triangular splitting of the saddle point matrix secondly we propose a modified block triangular splitting mbts this new preconditioner is easily implemented since it has simple block structure the spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed respectively moreover we apply the mgrs and the mbts preconditioners to three dimensional linearized navier stokes equations then we derive the quasi optimal parameters of the mgrs and the mbts preconditioners for two and three dimensional navier stokes equations respectively finally numerical experiments are illustrated to show the preconditioning effects of the two new preconditioners

a fractional variational iteration method for solving fractional nonlinear differential equations recently fractional differential equations have been investigated by employing the famous variational iteration method however all the previous works avoid the fractional order term and only handle it as a restricted variation a fractional variational iteration method was first proposed in g c wu e w m lee fractional variational iteration method and its application phys lett a 374 2010 2506 2509 and gave a generalized lagrange multiplier in this paper two fractional differential equations are approximately solved with the fractional variational iteration method

stochastic calculus for fractional brownian motion i theory in this paper a stochastic calculus is given for the fractional brownian motions that have the hurst parameter in 1 2 1 a stochastic integral of ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment this integral uses the wick product and a derivative in the path space some ito formulae or change of variables formulae are given for smooth functions of a fractional brownian motion or some processes related to a fractional brownian motion a stochastic integral of stratonovich type is defined and the two types of stochastic integrals are explicitly related a square integrable functional of a fractional brownian motion is expressed as an infinite series of orthogonal multiple integrals

a relaxed upper and lower triangular splitting preconditioner for the linearized navier stokes equation abstract based on the block triangular splitting of the original coefficient matrix a relaxed upper and lower triangular splitting rult preconditioner for the linearized incompressible navier stokes equations is considered the convergence analysis of the rult iteration method is presented the spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed respectively then the quasi optimal parameters of the rult preconditioner are derived finally some numerical results are carried out to show the effectiveness of the rult preconditioner and verify that the rult preconditioner outperforms the mds grs mrs and rbts preconditioners for the gmres method for solving the linearized incompressible navier stokes equation in terms of the number of iterations and computation times

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

a variant of relaxed triangular splitting preconditioners for generalized saddle point problems from navier stokes equations abstract based on the dimensional split preconditioner in this paper a variant of relaxed triangular splitting preconditioner vrts is presented and discussed in which a simple and feasible way is designed for the selection of the parameters spectral properties of the vrts preconditioned matrix are analyzed in detail theoretical analysis shows that when applying the vrts preconditioner within a krylov subspace method less computational cost is required at each iteration than some state of the art approaches finally some numerical experiments arising from the discretization of navier stokes equations are given to illustrate and validate the efficiency and robustness of the presented preconditioners with using gmres as an iterative solver

analytical treatment of differential equations with fractional coordinate derivatives this paper outlines a reliable strategy to use the homotopy perturbation method based on jumarie s derivative for solving fractional differential equations in this framework compact structures of fourth order fractional diffusion wave equations are considered as prototype examples moreover convergence of the proposed approach for these types of equations is investigated results show that the response expressions are mittag leffler stable

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

idea of invariant subspace combined with elementary integral method for investigating exact solutions of time fractional npdes abstract in this paper inspired by the idea of invariant subspace method and combined with elementary integral method we introduced a novel approach for investigating exact solutions of a time fractional nonlinear partial differential equation npde based on hypothetical structure of solution of separated variable a time fractional npde defined by time and space variables can be reduced to a nonlinear ordinary differential equation node or nodes defined by space variable alone and then using the elementary integral method to solve the node or nodes different kinds of exact solutions of a time fractional npde are obtained finally as examples the time fractional hunter saxton equation and time fractional li olver equation were studied different kinds of exact solutions of these equations were obtained and their dynamical properties were illustrated

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

relaxed block upper lower triangular preconditioner for generalized saddle point problems from the incompressible navier stokes equations abstract based on a new matrix splitting of the original coefficient matrix a relaxed block upper lower triangular rbult preconditioner is proposed for the solution of generalized saddle point problem arising from the incompressible navier stokes equations the distinct advantage of this preconditioner is that it is computationally cheaper to implement than the modified dimensional split mds preconditioner the generalized relaxed splitting grs preconditioner and the modified relaxed splitting mrs preconditioner when using krylov subspace methods the eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are given finally numerical results are provided to demonstrate the validity of the theoretical analysis which indicate not only that the rbult preconditioner is competitive when compared with other state of the art preconditioners but that it is more effective

a review of definitions of fractional derivatives and other operators abstract given the increasing number of proposals and definitions of operators in the scope of fractional calculus it is important to introduce a systematic classification nonetheless many of the definitions that emerged in the literature can not be considered as fractional derivatives we analyze a list of expressions to have a general overview of the concept of fractional integrals derivatives moreover some formulae that do not involve the term fractional are also included due to their particular interest in the area

on parameterized block triangular preconditioners for generalized saddle point problems in this paper we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems these preconditioners generalize the common block diagonal and triangular preconditioners we will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues numerical experiments of a model stokes problem are presented

induction of multiple fuzzy decision trees based on rough set technique the integration of fuzzy sets and rough sets can lead to a hybrid soft computing technique which has been applied successfully to many fields such as machine learning pattern recognition and image processing the key to this soft computing technique is how to set up and make use of the fuzzy attribute reduct in fuzzy rough set theory given a fuzzy information system we may find many fuzzy attribute reducts and each of them can have different contributions to decision making if only one of the fuzzy attribute reducts which may be the most important one is selected to induce decision rules some useful information hidden in the other reducts for the decision making will be losing unavoidably to sufficiently make use of the information provided by every individual fuzzy attribute reduct in a fuzzy information system this paper presents a novel induction of multiple fuzzy decision trees based on rough set technique the induction consists of three stages first several fuzzy attribute reducts are found by a similarity based approach and then a fuzzy decision tree for each fuzzy attribute reduct is generated according to the fuzzy id3 algorithm the fuzzy integral is finally considered as a fusion tool to integrate the generated decision trees which combines together all outputs of the multiple fuzzy decision trees and forms the final decision result an illustration is given to show the proposed fusion scheme a numerical experiment on real data indicates that the proposed multiple tree induction is superior to the single tree induction based on the individual reduct or on the entire feature set for learning problems with many attributes

transport and its enhancement caused by coupling using the weiss mean field approximation theory and the particles transport theory for the spatially periodic stochastic systems we derive an exact analytical expression for the stationary probability current of a coupled lattice system driven by dichotomous noise it is shown that for this coupled lattice system the spatial asymmetry of the system the asymmetry of the dichotomous noise and the coupling among nearest neighbors are the ingredients for the stationary probability current by applying our theory to two special models we find that 1 the coupling can lead to the directional transport of the particles even when the potential and the dichotomous noise are symmetric and 2 the coupling among nearest neighbors can enhance the transport of the particles in some circumstances our results are applied to a device of two dimensional josephson junction arrays and a large protein motors cluster

a class of new extended shift splitting preconditioners for saddle point problems abstract in this paper a class of new extended shift splitting ness preconditioners containing t 0 are proposed for saddle point problems which contain the existing shift splitting based preconditioners as special cases the convergence properties of the ness iteration and the spectral distribution of the ness preconditioned matrix are investigated we give an estimated bound for the real eigenvalues of the ness preconditioned matrix whose left end is positive and show that the non real eigenvalues of the ness preconditioned matrix are located in an intersection of two rings particularly these non real eigenvalues are located in an intersection of two rings and one circle if t 1 2 moreover we show that under appropriate conditions our proposed eigenvalue bounds are tighter than the corresponding bounds given in ren et al 2017 using fourier analysis we derive an optimized parameter t independent of the viscosity for the continuous version of the ness preconditioned gmres method for the 2d stokes equation moreover we find that the ness preconditioned gmres method with a constant multiple of the optimized parameter t is effective and robust to solve 2d stokes problems in practice numerical experiments are used to verify our theoretical results

stochastic resonance in biological nonlinear evolution models we investigate stochastic resonance in the nonlinear one dimensional fisher eigen model fem which represents an archetypal model for biological evolution based on a global coupling scheme in doing so we consider different periodically driven fitness functions which govern the evolution of a biological phenotype population for the case of a simple harmonic fitness function we are able to derive the exact analytic solution for the asymptotic probability density a distinct feature of this solution is a phase lag between the driving signal and the linear response of the system furthermore for more complex systems a general perturbation theory linear response approximation is put forward using the latter approach we investigate stochastic resonance in terms of the spectral amplification measure for a quadratic a quartic single peaked and for a bistable fitness function our analytical results are also compared with those of detailed numerical simulations our findings vindicate that stochastic resonance does occur in these nonlinear globally coupled biological systems

different methods for 3 1 dimensional space time fractional modified kdv zakharov kuznetsov equation in this paper the fractional derivatives in the sense of modified riemann liouville derivative and the exp function method the g g expansion method and the generalized kudryashov method are used to construct exact solutions for 3 1 dimensional space time fractional modified kdv zakharov kuznetsov equation this fractional equation can be turned into another nonlinear ordinary differential equation by fractional complex transformation and then these three methods are applied to solve it as a result some new exact solutions are obtained the three methods demonstrate power reliability and efficiency

transitions and transport for a spatially periodic stochastic system with locally coupled oscillators in this paper with a special model we investigate the spatially periodic stochastic system with locally coupled oscillators subject to a constant force f a nonequilibrium second order phase transition is found when f 0 this phase transition is reentrant when the additive noise is weak with varying the constant force f a continuous or discontinuous transition between the states with positive and negative mean fields m 0 and m 0 is observed which is not a phase transition the mean field or current sometimes exhibits hysteresis as a function of f with the variation of the force f when hysteresis of the mean field or current versus f appears a nonzero probability current with definite direction will occur at the point f 0 the correlation between the additive and multiplicative noises has an effect on the transitions and the transport

the first integral method for some time fractional differential equations abstract in this paper the fractional derivatives in the sense of modified riemann liouville derivative and the first integral method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations the power of this manageable method is presented by applying it to several examples this approach can also be applied to other nonlinear fractional differential equations

lie symmetry analysis of the time fractional kdv type equation abstract the lie symmetry analysis method is extended to deal with the time fractional kdv type equation it is shown that this equation can be reduced to an equation with the erdelyi kober fractional derivative this method can also be applied to symmetry classification of fractional equations with some arbitrary functions

hermitian and normal splitting methods for non hermitian positive definite linear systems in this paper a hermitian and normal splitting hns iterative method is proposed for solving non hermitian positive definite linear systems arising from convection diffusion reaction equations theoretical analysis shows that the hns method converges unconditionally to the exact solution of the linear system choices of the parameters are discussed the corresponding inexact hns ihns iterative method which is developed by employing some krylov subspace methods as its inner process is studied finally some numerical experiments are presented to demonstrate the effectiveness of the new method

a fractional order adaptive regularization primal dual algorithm for image denoising this paper aims to develop a fractional order model and a primal dual algorithm for image denoising where a regularization parameter can be adjusted adaptively according to morozov discrepancy principle at each iteration to ensure that the denoised image retains in a specific set in the light of saddle point theory the convergence of our proposed algorithm is guaranteed simulations with comparisons are carried out to demonstrate the effectiveness of our proposed algorithm for image denoising

new choices of preconditioning matrices for generalized inexact parameterized iterative methods for large sparse saddle point problems chen and jiang recently studied a class of generalized inexact parameterized iterative methods see f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 in this paper the methods are modified and some choices of preconditioning matrices are given these preconditioning matrices have advantages in solving large sparse linear system numerical experiments of a model stokes problem are presented

a fokker planck equation of fractional order with respect to time by combining the maximum entropy principle with some considerations related to derivatives of fractional order one is led to suggest a fokker planck of fractional order with respect to time which could be related to dynamical systems subject to fractional brownian motion the relation with the process associated with the equation p t 1 n 1 2np x2n is exhibited

a note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices in this note we consider a set of augmentation block preconditioners for solving generalized saddle point systems whose coefficient matrices have singular 1 1 blocks results concerning the eigenvalue distribution and forms of the eigenvectors of the augmentation block preconditioned generalized saddle point matrix and its minimal polynomial are given and an optimal augmentation block preconditioner of the set is derived these results extend previous ones in the literature numerical experiments that show the very effective performance of the optimal augmentation block preconditioner are reported

cross fuzzy entropy a new method to test pattern synchrony of bivariate time series a new method namely cross fuzzy entropy c fuzzyen analysis that can enable the measurement of the synchrony or similarity of patterns between two distinct signals is presented in this study with the inclusion of fuzzy sets the similarity of vectors is fuzzily defined in c fuzzyen based on the exponential function and their shapes rather than on the heaviside function used in the conventional cross sample entropy c sampen tests on simulated data sets and real eeg signals showed that c fuzzyen was superior to c sampen in several aspects including giving the entropy definition in the case of small parameters better relative consistency and less dependence on record length the proposed c fuzzyen was then applied for the analysis of simultaneously recorded electromyography emg and mechanomyography mmg signals during sustained isometric contraction for monitoring local muscle fatigue the results showed that the c fuzzyen of emg mmg signals decreased significantly during the development of muscle fatigue the c fuzzyen showed a similar trend with the mean frequency mnf of emg the commonly used muscle fatigue indicator however c fuzzyen of emg mmg demonstrated a better robustness to the length of the analysis window in comparison with the mnf of emg the results suggested that the proposed c fuzzyen of emg mmg may potentially become a new reliable method for muscle fatigue assessment it can also be applied to other bivariate signals extracted from complex systems with short data lengths in noisy backgrounds