what is the fractional laplacian a comparative review with new results abstract the fractional laplacian in r d which we write as 2 with 0 2 has multiple equivalent characterizations moreover in bounded domains boundary conditions must be incorporated in these characterizations in mathematically distinct ways and there is currently no consensus in the literature as to which definition of the fractional laplacian in bounded domains is most appropriate for a given application the riesz or integral definition for example admits a nonlocal boundary condition where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional laplacian in contrast the spectral definition requires only the standard local boundary condition these differences among others lead us to ask the question what is the fractional laplacian beginning from first principles we compare several commonly used definitions of the fractional laplacian theoretically through their stochastic interpretations as well as their analytical properties then we present quantitative comparisons using a sample of state of the art methods we discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems radial basis function collocation for the riesz fractional laplacian and nonharmonic lifting for the spectral fractional laplacian in our numerical studies we aim to compare different definitions on bounded domains using a collection of benchmark problems we consider the fractional poisson equation with both zero and nonzero boundary conditions where the fractional laplacian is defined according to the riesz definition the spectral definition the directional definition and the horizon based nonlocal definition we verify the accuracy of the numerical methods used in the approximations for each operator and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions through our efforts we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications

generalized means as model of compensative connectives abstract empirical investigations have shown that instead of strong logical and and or connectives there is often some degree of compensation in between in this paper the generalized means are proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation in a natural manner including the arithmetic and geometric means as well as the maximum and minimum operators as special cases numerical examples with empirical data form an illustration of this concept some aspects of applications to decision making are discussed

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

ruin probability with claims modeled by a stationary ergodic stable process for a random walk with negative drift we study the exceedance probability ruin probability of a high threshold the steps of this walk claim sizes constitute a stationary ergodic stable process we study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes our findings show that the order of magnitude of the ruin probability varies significantly from one model to another in particular ruin becomes much more likely when the claim sizes exhibit long range dependence the proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a poisson process

error estimates of a spectral petrov galerkin method for two sided fractional reaction diffusion equations abstract we study regularity and the spectral method for two sided fractional diffusion equations with a reaction term we show that the regularity of the solution in weighted sobolev spaces can be greatly improved compared to that in standard sobolev spaces with this regularity we prove an optimal error estimate for the spectral petrov galerkin method numerical results are presented to verify our theoretical convergence orders

sample path properties of self similar processes with stationary increments a real valued process x x t u is self similar with exponent h h ss if x a d ahx for all a 0 sample path properties of h ss processes with stationary increments are investigated the main result is that the sample paths have nowhere bounded variation if 0 1 in which case they are singular however nowhere bounded variation may occur also for h 1 examples exhibiting this combination of properties are constructed as well as many others most are obtained by subordination of random measures to point processes in ir2 that are poincar6 i e invariant in distribution for the transformations t x at b ax of r in a final section it is shown that the self similarity and stationary increment properties are preserved under composition of independent processes x1 a x2 x1 x2 t l some interesting examples are obtained this way

multi domain spectral collocation method for variable order nonlinear fractional differential equations abstract spectral and spectral element methods using galerkin type formulations are efficient for solving linear fractional pdes fpdes of constant order but are not efficient in solving nonlinear fpdes and cannot handle fpdes with variable order in this paper we present a multi domain spectral collocation method that addresses these limitations we consider fpdes in the riemann liouville sense and employ jacobi lagrangian interpolants to represent the solution in each element we provide variable order differentiation formulas which can be computed efficiently for the multi domain discretization taking into account the nonlocal interactions we enforce the interface continuity conditions by matching the solution values at the element boundaries via the lagrangian interpolants and in addition we minimize the jump in integer fluxes using a penalty method we analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme we demonstrate the effectiveness of the new method for the fractional helmholtz equation of constant and variable order using h p refinement for different values of the penalty parameter we also solve the fractional burgers equation with constant and variable order and compare with solutions obtained with a single domain spectral collocation method

a generalized spectral collocation method with tunable accuracy for variable order fractional differential equations we generalize existing jacobi gauss lobatto collocation methods for variable order fractional differential equations using a singular approximation basis in terms of weighted jacobi polynomials of the form 1 pm x mu p j a b x where mu 1 in order to derive the differentiation matrices of the variable order fractional derivatives we develop a three term recurrence relation for both integrals and derivatives of these weighted jacobi polynomials hence extending the three term recurrence relationship of jacobi polynomials the new spectral collocation method is applied to solve fractional ordinary and partial differential equations with endpoint singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solution by properly tuning the parameter mu even for cases where we do not know explicitly the form of singularity in the solution at the boundaries

fractional sturm liouville eigen problems we first consider a regular fractional sturm liouville problem of two kinds rfslp i and rfslp ii of order 0 2 the corresponding fractional differential operators in these problems are both of riemann liouville and caputo type of the same fractional order 2 0 1 we obtain the analytical eigensolutions to rfslp i ii as non polynomial functions which we define as jacobi poly fractonomials these eigenfunctions are orthogonal with respect to the weight function associated with rfslp i ii subsequently we extend the fractional operators to a new family of singular fractional sturm liouville problems of two kinds sfslp i and sfslp ii we show that the primary regular boundary value problems rfslp i ii are indeed asymptotic cases for the singular counterparts sfslp i ii furthermore we prove that the eigenvalues of the singular problems are real valued and the corresponding eigenfunctions are orthogonal in addition we obtain the eigen solutions to sfslp i ii analytically also as non polynomial functions hence completing the whole family of the jacobi poly fractonomials in numerical examples we employ the new poly fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results

weighted sums of i i d random variables attracted to integrals of stable processes under the assumption of the convergence of partial sum processesz n t of i i d random variables to an stable levy processz t 0 2 the convergence of weighted sums f n u dz n u to f u dz u is studied the general convergence result is then applied to examine the domain of attraction of the fractional stable process

stable bounded subsets of lα and sample unboundedness of symmetric stable processes let xt be the levy symmetric stable process with index 1 2 and let c be a convex symmetric subset of l 0 1 we prove that if c does not have compact closure then sup c 01 t dxt is infinite with probability one this extends a result of dudley in the case 2

least squares finite element solution of a fractional order two point boundary value problem in this paper a theoretical framework for the least squares finite element approximation of a fractional order differential equation is presented mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established using these properties existence and uniqueness of the least squares approximation is proven optimal error estimates are proven for piecewise linear trial elements numerical results are included which confirm the theoretical results

a fast finite volume method for conservative space time fractional diffusion equations discretized on space time locally refined meshes abstract a fast finite volume method was developed for conservative space time fractional diffusion partial differential equations in two space dimensions in the method a locally refined composite spatial mesh is used to resolve the boundary layers of the solutions and a graded temporal mesh is used to resolve the singularity of the solutions near the initial time a fast numerical method was developed by fully exploring the structure of the stiffness matrices numerical results show the utility of the method

implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear riesz space fractional diffusion equation abstract a numerical method with high accuracy both in time and in space is proposed for the two dimensional nonlinear riesz space fractional diffusion equation the main idea is based on a spectral galerkin method in spatial direction and an s stage implicit runge kutta method in temporal direction a rigorous stability and error analysis is performed for the proposed method it is shown that the proposed method is stable and convergent the optimal spatial error estimate is also derived numerical experiments are provided to illustrate the theoretical results

spectral approximation of a variable coefficient fractional diffusion equation in one space dimension abstract in this article we consider the approximation of a variable coefficient two sided fractional diffusion equation fde having unknown u by introducing an intermediate unknown q the variable coefficient fde is rewritten as a lower order constant coefficient fde a spectral approximation scheme using jacobi polynomials is presented for the approximation of q q n the approximate solution to u u n is obtained by post processing q n an a priori error analysis is given for q q n and u u n two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates

crank nicolson galerkin spectral method for solving two dimensional time space distributed order weakly singular integro partial differential equation abstract the fractional pdes based upon the distributed order fractional derivative have several applications in physics the two dimensional time space distributed order weakly singular integro partial differential model is investigated by a combination of finite difference and galerkin spectral methods a second order finite difference formula is employed to approximate the temporal variable in this stage the stability and convergence of the semi discrete scheme are proved then the galerkin spectral method based on the modified jacobi polynomials is applied to discrete the space variable also in this step the error estimate of the full discrete scheme is studied finally two test problems have been presented to confirm the theoretical results

boundary conditions for two sided fractional diffusion abstract this paper develops appropriate boundary conditions for the two sided fractional diffusion equation where the usual second derivative in space is replaced by a weighted average of positive left and negative right fractional derivatives mass preserving reflecting boundary conditions for two sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary stable consistent explicit and implicit euler methods are detailed and steady state solutions are derived steady state solutions for two sided fractional diffusion equations using both riemann liouville and caputo flux are computed for riemann liouville flux and reflecting boundary conditions the steady state solution is singular at one or both of the end points for caputo flux and reflecting boundary conditions the steady state solution is a constant function numerical experiments illustrate the convergence of these numerical methods finally the influence of the reflecting boundary on the steady state behavior subject to both the riemann liouville and caputo fluxes is discussed

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

a self similar process with nowhere bounded sample paths on considere un processus auto similaire a parametre h et on montre la non existence de trajectoires bornees pour certaines valeurs de et

a petrov galerkin finite element method for fractional convection diffusion equations in this work we develop variational formulations of petrov galerkin type for one dimensional fractional boundary value problems involving either a riemann liouville or caputo derivative of order alpha in 3 2 2 in the leading term and both convection and potential terms they arise in the mathematical modeling of asymmetric superdiffusion processes in heterogeneous media the well posedness of the formulations and sharp regularity pickup of the variational solutions are established a novel finite element method fem is developed which employs continuous piecewise linear finite elements and shifted fractional powers for the trial and test space respectively the new approach has a number of distinct features it allows the derivation of optimal error estimates in both the l 2 d and h 1 d norms and on a uniform mesh the stiffness matrix of the leading term is diagonal and the resulting linear system is well conditioned further in the riemann liouville case an enriched fem is propose

adaptive finite element method for fractional differential equations using hierarchical matrices abstract a robust and fast solver for the fractional differential equation fdes involving the riesz fractional derivative is developed using an adaptive finite element method it is based on the utilization of hierarchical matrices h matrices for the representation of the stiffness matrix resulting from the finite element discretization of the fdes we employ a geometric multigrid method for the solution of the algebraic system of equations we combine it with an adaptive algorithm based on a posteriori error estimation a posteriori error estimation based adaptive algorithm is used to deal with general type singularities arising in the solution of the fdes through various test examples we demonstrate the efficiency of the method and the high accuracy of the numerical solution even in the presence of singularities the proposed technique has been verified effectively through fundamental examples including riesz left right riemann liouville fractional derivative and furthermore it can be readily extended to more general fractional differential equations with different boundary conditions and low order terms

hierarchical genetic fuzzy systems abstract this paper introduces a hierarchical evolutionary approach to optimize the parameters of takagi sugeno ts fuzzy systems the approach includes a least squares method to determine the parameters of nonlinear consequents a pruning procedure is developed to avoid redundancy in each rule consequent and to achieve proper representation flexibility the performance of the hierarchical evolutionary approach is evaluated using function approximation and classification problems they demonstrate that the evolutionary algorithm working together with optimization and pruning procedures provides structurally simple fuzzy systems whose performance seems to be better than the ones produced by alternative approaches

a generalized spectral collocation method with tunable accuracy for fractional differential equations with end point singularities we develop spectral collocation methods for fractional differential equations with variable order with two end point singularities specifically we derive three term recurrence relations for both integrals and derivatives of the weighted jacobi polynomials of the form 1 x mu 1 1 x mu 2 p j a b x a b mu 1 mu 2 1 which leads to the desired differentiation matrices we apply the new differentiation matrices to construct collocation methods to solve fractional boundary value problems and fractional partial differential equations with two end point singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solutions by properly tuning the parameters mu 1 and mu 2 even for cases for which we do not know explicitly the form of singularities in the solution at the boundaries comparison with other existing methods shows the superior accuracy of the proposed spectral collocation method

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

particle swarm optimization for determining fuzzy measures from data fuzzy measures and fuzzy integrals have been successfully used in many real applications how to determine fuzzy measures is the most difficult problem in these applications though there have existed some methodologies for solving this problem such as genetic algorithms gradient descent algorithms and neural networks it is hard to say which one is more appropriate and more feasible each method has its advantages and limitations therefore it is necessary to develop new methods or techniques to learn distinct fuzzy measures in this paper we make the first attempt to design a special particle swarm algorithm to determine a type of general fuzzy measures from data and demonstrate that the algorithm is effective and efficient furthermore we extend this algorithm to identify and revise other types of fuzzy measures to test our algorithms we compare them with the basic particle swarm algorithms gradient descent algorithms and genetic algorithms in literatures in addition for verifying whether our algorithms are robust in noisy situations a number of numerical experiments are conducted theoretical analysis and experimental results show that for determining fuzzy measures the particle swarm optimization is feasible and has a better performance than the existing genetic algorithms and gradient descent algorithms

qualitative decision theory with preference relations and comparative uncertainty an axiomatic approach this paper investigates a purely qualitative approach to decision making under uncertainty since the pioneering work of savage most models of decision under uncertainty rely on a numerical representation where utility and uncertainty are commensurate giving up this tradition we relax this assumption and introduce an axiom of ordinal invariance requiring that the decision maker s preference between two acts only depends on the relative position of their consequences for each state within this qualitative framework we determine the only possible form of the corresponding decision rule then assuming the transitivity of the strict preference the underlying partial confidence relations are those at work in non monotonic inference and thus satisfy one of the main properties of possibility theory the satisfaction of additional postulates of unanimity and anonymity enforces the use of a necessity measure unique up to a monotonic transformation for encoding the relative likelihood of events

a crank nicolson adi galerkin legendre spectral method for the two dimensional riesz space distributed order advection diffusion equation abstract in the paper a crank nicolson alternating direction implicit adi galerkin legendre spectral scheme is presented for the two dimensional riesz space distributed order advection diffusion equation the gauss quadrature has a higher computational accuracy than the mid point quadrature rule which is proposed to approximate the distributed order riesz space derivative so that the considered equation is transformed into a multi term fractional equation moreover the transformed equation is solved by discretizing in space by the adi galerkin legendre spectral scheme and in time using the crank nicolson difference method stability and convergence analysis are verified for the numerical approximation a lot of numerical results are demonstrated to justify the theoretical analysis

a spectral method of exponential convergence for singular solutions of the diffusion equation with general two sided fractional derivative an open problem in the numerical solution of fractional partial differential equations fpdes is how to obtain high order accuracy for singular solutions even for smooth right hand sides solutions of fpdes are singular here we consider the one dimensional diffusion equation with general two sided fractional derivative characterized by a parameter p in 0 1 for p 1 2 we recover the riesz fractional derivative while for p 1 0 we obtain the one sided fractional derivative we employ a petrov galerkin projection in a properly weighted sobolev space with two sided jacobi polyfracnomials as basis and test functions in particular we derive these two sided jacobi polyfractonomials as eigenfunctions of a sturm liouville problem with weights uniquely determined by the parameter p we provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term i e for smooth data corresponding to singular solutions we obtain exponential convergence wh

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

k order additive discrete fuzzy measures and their representation abstract in order to face with the complexity of discrete fuzzy measures we propose the concept of k orderadditive fuzzy measure including usual additive measures and fuzzy measures every discrete fuzzy measure is a k order additive fuzzy measure for a unique k a related topic of the paper is to introduce an alternative representation of fuzzy measures called the interaction representation which sets and extends in a common framework the shapley value and the interaction index proposed by murofushi and soneda

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

on set valued fuzzy measures so far there have been many discussions on sugeno s fuzzy measures and fuzzy integrals but most of them are concentrated on single valued functions motivated by aumann s set valued integral j math anal appl 12 1965 1 we have introduced the fuzzy integral of set valued functions fuzzy sets syst 56 1993 237 fuzzy sets syst 76 1995 365 fuzzy sets syst 78 1996 341 ph d dissertation harbin institute of technology 1998 it is a well behaved extension of fuzzy integrals of single valued functions it is well known that arstein s set valued measure trans am math soc 165 1972 103 is an important branch in set valued analysis set valued analysis birkhauser berlin 1990 or theory of correspondences theory of correspondences wiley new york 1984 compared to it the present paper will try to establish the basic idea of set valued fuzzy measures which are monotone set valued set functions it is also a natural generalization of single valued fuzzy measures as well as an extension of set valued measures in the case of one dimension these works include the concept of set valued fuzzy measures set valued fuzzy measures defined by set valued fuzzy integrals and set valued pseudo additive measures it can be viewed as a continuation of previous work fuzzy sets syst 56 1993 237 fuzzy sets syst 76 1995 365 fuzzy sets syst 78 1996 341

efficient spectral galerkin methods for fractional partial differential equations with variable coefficients efficient spectral galerkin algorithms are developed to solve multi dimensional fractional elliptic equations with variable coefficients in conserved form as well as non conserved form these algorithms are extensions of the spectral galerkin algorithms for usual elliptic pdes developed in 24 more precisely for separable fpdes we construct a direct method by using a matrix diagonalization approach while for non separable fpdes we employ a preconditioned bicgstab method with a suitable separable fpde with constant coefficients as preconditioner the cost of these algorithms is of o n d 1 flops where d is the space dimension we derive rigorous weighted error estimates which provide more precise convergence rate for problems with singularities at boundaries we also present ample numerical results to validate the algorithms and error estimates

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

aggregation functions means this two part state of the art overview on aggregation theory summarizes the essential information concerning aggregation issues an overview of aggregation properties is given including the basic classification on aggregation functions in this first part the stress is put on means i e averaging aggregation functions both with fixed arity n ary means and with multiple arities extended means

a hierarchical model of a linguistic variable in this work a theoretical hierarchical model of dichotomous linguistic variables is presented the model incorporates certain features of the approximate reasoning approach and others of the fuzzy control approach to fuzzy linguistic variables it allows sharing of the same hierarchical structure between the syntactic definition of a linguistic variable and its semantic definition given by fuzzy sets this fact makes it easier to build symbolic operations between linguistic terms with a better grounded semantic interpretation moreover the family of fuzzy sets which gives the semantics of each linguistic term constitutes a multiresolution system and thanks to that any fuzzy set can be represented in terms of the set of linguistic terms the model can also be considered a general framework for building more interpretable fuzzy linguistic variables with a high capacity of accuracy which could be used to build more interpretable fuzzy rule based systems frbs

numerical solution of fractional integro differential equations by collocation method abstract this paper deals with the numerical solution of fractional integro differential equations of the type d q y t p t y t f t 0 t k t s y s d s t i 0 1 by polynomial spline functions we derive a system of equations that characterizing the numerical solution some numerical examples are also provided to illustrated our results

bounded variation double sequence space of fuzzy real numbers in this paper we have introduced the notion of fuzzy real valued bounded variation double sequence space b2v f we have studied some of its properties like convergence free solidness symmetricity monotonicity etc we have proved some inclusion results too

an interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy measure abstract in this paper the non additivity of sugeno s fuzzy measures is interpreted in terms of addition and the rationality of the choquet integral is discussed it is pointed out that a fuzzy measure on a set x expresses the interaction between the subsets of x and can be represented by an additive measure it is shown through concrete examples that the choquet integral is reasonable as an integral with respect to a fuzzy measure it is also found that the choquet integral is closely related with the representation of a fuzzy measure

a high order schema for the numerical solution of the fractional ordinary differential equations in this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations fodes this technique is based on the so called block by block approach which is a common method for the integral equations in our approach the classical block by block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps while preserving the good stability property of the block by block schemes by using this new approach we are able to construct a high order schema for fodes of the order a a 0 the stability and convergence of the schema is rigorously established we prove that the numerical solution converges to the exact solution with order 3 a for 0 1 a series of numerical examples are provided to support the theoretical claims

sharp error bounds for jacobi expansions and gegenbauer gauss quadrature of analytic functions this paper provides a rigorous and delicate analysis for exponential decay of jacobi polynomial expansions of analytic functions associated with the bernstein ellipse using an argument that can recover the best estimate for the chebyshev expansion we derive various new and sharp bounds of the expansion coefficients which are featured with explicit dependence of all related parameters and valid for degree n ge 1 we demonstrate the sharpness of the estimates by comparing with existing ones in particular the very recent results in siam j numer anal 50 2012 pp 1240 1263 we also extend this argument to estimate the gegenbauer gauss quadrature remainder of analytic functions which leads to some new tight bounds for quadrature errors

optimal convergence rates for semidiscrete finite element approximations of linear space fractional partial differential equations under minimal regularity assumptions abstract we consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by jin et al 2015 and ervin et al 2018 the error estimates are proved by using two approaches one approach is to apply the duality argument in johnson 1987 for the heat equation to consider the error estimates for the linear space fractional partial differential equations this argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution another approach is to use the approximate solution operators of the corresponding fractional elliptic problems this argument can be extended to consider more general linear space fractional partial differential equations numerical examples are given to show that the numerical results are consistent with the theoretical results

complex structured decision making model a hierarchical frame work for complex structured data we introduce a hierarchical framework we call complex structured decision making model for complexly structured knowledge representation in intelligent decision making we show that our model extends non hierarchical flat decision making models to hierarchical decision making models that are similar to comprehensible human decision making processes further we make an argument that hierarchial representation of knowledge in a complex structured decision making model simplifies the approximation of aggregation functions to easily adapt to the underline relation of the system additionally using a real world complex structured data set we show that hierarchically organized fuzzy integrals e g choquet integral and sugeno integral and fuzzy signatures outperform these non hierarchical fuzzy integrals

a fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation abstract we develop a fast solver for the spectral element method sem applied to the two sided fractional diffusion equation on uniform geometric and graded meshes by approximating the singular kernel with a degenerate kernel we construct a hierarchical matrix h matrix to represent the stiffness matrix of the sem and provide error estimates which we verify numerically we can solve efficiently the h matrix approximation problem using a hierarchical lu h lu decomposition method which reduces the computational cost to o r 2 n d log 2 n o r 3 n d log n where r is the rank of the h lu decomposition n d is the total number of degrees of freedom and n is the number of elements however we lose the high accuracy of the sem to prevent this we solve the corresponding preconditioned system by using the h matrix approximation problem as a preconditioner hence recovering the high order accuracy of the sem numerical results show that the condition number of the preconditioned system is independent of the polynomial degree p and grows with the number of elements but at modest values of the rank r here r is the rank of submatrices of the h matrix approximation it is below order 10 in our experiments which represents a reduction of more than 11 orders of magnitude from the unpreconditioned system this reduction is higher in the two sided fractional derivative compared to one sided fractional derivative the corresponding cost is o r 2 n d log 2 n o r 3 n d log n o n d 2 moreover by using a structured mesh uniform or geometric mesh we can further reduce the computational cost to o r 2 n d log 2 n o r 3 n d log n o p 2 n log n for the preconditioned system furthermore we compare the present method with the adaptive cross approximation aca which is kernel independent and relies on the coefficient matrix the comparison shows that the accuracy using the present algorithm is better than the one using aca for the h matrix approximation problem and furthermore we have observed that the aca method may fail to converge in some cases