uncertainty measures for interval set information tables based on interval δ similarity relation abstract the notion of uncertainty measure is one of the most important topics in rough set theory and has been studied in different kinds of information tables however few studies have focused on the interval set information table which is regarded as one of the generalized models of single valued information tables this paper aims at studying the uncertainty measurements for interval set information tables firstly an interval similarity relation is induced based on the similarity degree the similarity relation induces the granules which form a covering in interval set information tables secondly four types of granularity measures are defined to measure the granularity of a covering thirdly the concepts of accuracy and roughness in rough set theory are respectively extended to accuracy and roughness for interval set information tables furthermore four new combinations ofuncertainty measures by considering proposed granularity measures and accuracy and roughness are defined and analyzed theoretical analyses and experimental results illustrate that the proposed measures are effective and accurate for interval set information tables

uncertainty measurement for interval valued information systems abstract interval valued information systems are generalized models of single valued information systems accuracy and roughness are employed to depict the uncertainty of a set under an attribute subset in a pawlak rough set model based on equivalence classes information theoretic measures of uncertainty for rough sets have also been proposed however there are few studies on uncertainty measurements for interval valued information systems this paper addresses the uncertainty measurement problem in interval valued information systems the concept of the similarity degree based on the possible degree is introduced consequently the similarity relation between two interval objects are constructed by a given similarity rate based on the similarity relation similarity classes are defined under this definition accuracy and roughness are given for interval valued information systems which are generalizations of the concepts accuracy and roughness for the equivalence relation based rough set model moreover an alternative uncertainty measure called the rough degree is proposed theoretical studies and numerical experiments show that the proposed measures are effective and suitable for interval valued information systems

a measurement theory view on the granularity of partitions measurement of granularity is one of the foundational issues in granular computing this paper investigates a class of measures of granularity of partitions the granularity of a set is defined by a strictly monotonic increasing transformation of the cardinality of the set the granularity of a partition is defined as the expected granularity of all blocks of the partition with respect to the probability distribution defined by the partition many existing measures of granularity are instances of the proposed class new measures of granularity of partitions are also introduced

fuzzy feature selection based on min max learning rule and extension matrix in many systems such as fuzzy neural network we often adopt the language labels such as large medium small etc to split the original feature into several fuzzy features in order to reduce the computation complexity of the system after the fuzzification of features the optimal fuzzy feature subset should be selected in this paper we propose a new heuristic algorithm where the criterion is based on min max learning rule and fuzzy extension matrix is designed as the search strategy the algorithm is proved in theory and has shown its high performance over several real world benchmark data sets

bdf like methods for nonlinear dynamic analysis we present several time integration algorithms of second order accuracy that are numerically simple and effective for nonlinear elastodynamic problems these algorithms are based on a general four step scheme that has a resemblance to the backward differentiation formulas we also present an extension to the composite strategy of the bathe method appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity and less dissipative members of each algorithm have been identified we demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions and test these algorithms with several three dimensional nonlinear elastodynamic problems involving large deformations and rotations employing st venant kirchhoff and compressible neo hookean hyperelastic material models these tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well known instability

interval ordered information systems interval information systems are generalized models of single valued information systems by introducing a dominance relation to interval information systems we propose a ranking approach for all objects based on dominance classes and establish a dominance based rough set approach which is mainly based on substitution of the indiscernibility relation by the dominance relation furthermore we discuss interval ordered decision tables and dominance rules to simplify knowledge representation and extract much simpler dominance rules we propose attribute reductions of interval ordered information systems and decision tables that eliminate only the information that are not essential from the viewpoint of the ordering of objects or dominance rules the approaches show how to simplify an interval ordered information system and find dominance rules directly from an interval ordered decision table these results will be helpful for decision making analysis in interval information systems

optimal error estimates in jacobi weighted sobolev spaces for polynomial approximations on the triangle spectral approximations on the triangle by orthogonal polynomials are studied in this paper optimal error estimates in weighted semi norms for both the l2 and h 1 0 orthogonal polynomial projections are established by using the generalized koornwinder polynomials and the properties of the sturm liouville operator on the triangle these results are then applied to derive error estimates for the spectral galerkin method for second and fourth order equations on the triangle the generalized koornwinder polynomials and approximation results developed in this paper will be useful for many other applications involving spectral and spectral element approximations in triangular domains

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

a triangular spectral element method applications to the incompressible navier stokes equations abstract encouraged by the success of spectral elements methods in computational fluid dynamics and p type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers we introduce here a complete formulation using a modal basis which has been implemented in a new code ne t r the new basis has the following properties jacobi polynomials of mixed weights semi orthogonality hierarchical structure generalized tensor warped product variable order and a new apex co ordinate system allowing automated integration with gaussian quadrature we have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations we use this notation to formulate the linear advection and helmholtz equations in an efficient manner and show that we recover the following properties well conditioned matrices asymptotic operation count of o n 3 scaling o n 2 of the spectral radius of the weak convective operator and exponential convergence using polynomials up to 40 having constructed and numerically analysed these equations we are then able to solve the incompressible navier stokes equations using a high order splitting scheme we demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the stokes and navier stokes problems

unstructured spectral element methods for simulation of turbulent flows in this paper we present a spectral element fourier algorithm for simulating incompressible turbulent flows in complex geometries using unstructured quadrilateral meshes to this end we compare two different interface formulations for extending the conforming spectral element method in order to allow for surgical mesh refinement and still retain spectral accuracy the zanolli iterative procedure and variational patching based on auxiliary mortar functions we present an interpretation of the original mortar element method as a patching scheme and develop direct and iterative solution techniques that make the method efficient for simulations of turbulent flows the properties of the new method are analyzed in detail by studying the eigenspectra of the advection and diffusion operators we then present numerical results that illustrate the flexibility as well as the exponential convergence of the new algorithm for nonconforming discretizations we conclude with simulation studies of the turbulent cylinder wake at re 1000 external flow and turbulent flow over riblets at re 3280 internal flow

a weak legendre collocation spectral method for the solution of the incompressible navier stokes equations in unstructured quadrilateral subdomains a weak legendre spectral method is developed for the solution of the primitive variable formulation of the unsteady incompressible navier stokes equations in general two dimensional and axisymmetric geometries a semi implicit projection method is utilized for the temporal approximation and for decoupling the velocity field from the pressure field a series of elliptic boundary value problems arises from the above procedure each of which is spatially discretized by a weak collocation method in multiple nonoverlapping subdomains in particular a modified variational formulation of the partial differential equations is presented which leads after discretization to a weak multidomain approximation of the corresponding problems a weak formalism for the influence matrix technique is also developed which is consistent with the spatial discretization scheme and successfully separate the equations for the internal nodes from the ones governing the interface unknowns a method of dealing with the singularity problem faced by the weak formulation at axisymmetric problems is proposed while a combination of direct methods is studied for tackling effectively the linear algebraic systems resulting from the full discretization exponential convergence is demonstrated for a plethora of stokes and navier stokes simulations

high order finite elements applied to the discrete boltzmann equation a discontinuous galerkin approach for solving the discrete boltzmann equation is presented allowing to compute approximate solutions for fluid flow problems based on a two dimensional high order finite element and an explicit euler time stepping scheme the d2q9 model is discretized and the results are compared to the exact solution of the navier stokes equation four numerical examples are considered including stationary and instationary problems with curved boundaries it is demonstrated that the proposed method allows to obtain the desired highly efficient exponential convergence copyright 2006 john wiley sons ltd

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

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

on the dual boundary conditions and the nonnegative square root of the second derivative operator abstract a dual relationship between the boundary conditions of the second derivative operator au u is established some dual properties determined by the dual boundary conditions are considered furthermore the singularities of the nonnegative square root of the above second derivative operator are studied our results show that although a 1 2 u is not u it is a harmonic conjugate of u in some sense

higher order triangular spectral element method with optimized cubature points for seismic wavefield modeling the mass lumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points known as cubature points to date such points are only known analytically in tensor domains such as quadrilateral or hexahedral elements thus the diagonal mass matrix spectral element method sem in non tensor domains always relies on numerically computed interpolation points or quadrature points however only the cubature points for degrees 1 to 6 are known which is the reason that we have developed a p norm based optimization algorithm to obtain higher order cubature points in this way we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9 the dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration simultaneously the lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties as a result such points provide both good interpolation properties and integration accuracy the courantfriedrichslewy cfl numbers are tabulated for the conventional fekete based triangular spectral element tsem the tsem with exact integration and the optimized cubature based tsem otsem a complementary study demonstrates the spectral convergence of the otsem a numerical example conducted on a half space model demonstrates that the otsem improves the accuracy by approximately one order of magnitude compared to the conventional fekete based tsem in particular the accuracy of the 7th order otsem is even higher than that of the 14th order fekete based tsem furthermore the otsem produces a result that can compete in accuracy with the quadrilateral sem qsem the high accuracy of the otsem is also tested with a non flat topography model in terms of computational efficiency the otsem is more efficient than the fekete based tsem although it is slightly costlier than the qsem when a comparable numerical accuracy is required higher order cubature points for degrees 7 to 9 are developed the effects of quadrature rule on the mass and stiffness matrices has been conducted the cubature points have always positive integration weights freeing from the inversion of a wide bandwidth mass matrix the accuracy of the tsem has been improved in about one order of magnitude

spectral schemes on triangular elements abstract the poisson problem with homogeneous dirichlet boundary conditions is considered on a triangle the transformation from square to triangle is realized by mapping an edge of the square onto a corner of the triangle then standard chebyshev collocation techniques can be implemented numerical experiments demonstrate the expected high spectral accuracy for smooth solutions furthermore it is shown that finite difference preconditioning can be successfully employed to construct an efficient iterative solver then the convection diffusion equation is considered here finite difference preconditioning with central differences leads to instability however using the first order upstream scheme we obtain a stable method finally a domain decomposition technique is applied to the patching of rectangular and triangular elements

rough approximation quality revisited in rough set theory the approximation quality is the traditional measure to evaluate the classification success of attributes in terms of a numerical evaluation of the dependency properties generated by these attributes in this paper we re interpret the classical in terms of a classic measure based on sets the marczewski steinhaus metric and also in terms of proportional reduction of errors pre measures we also exhibit infinitely many possibilities to define like statistics which are meaningful in situations different from the classical one and provide tools to ascertain the statistical significance of the proposed measures which are valid for any kind of sample copyright 2001 elsevier science b v

information theoretic measures of uncertainty for rough sets and rough relational databases rough set theory has become well established as a mechanism for uncertainty management in a wide variety of applications including databases this paper addresses the measurement of uncertainty in rough sets and rough relational databases by introducing a measurement based on information theory this rough entropy is discussed as it applies to rough sets in general and in particular to aspects of the rough relational database model

rough set approach to incomplete numerical data rough set theory has been applied successfully in many fields however classical rough set model can only deal with complete and symbolic data sets some researchers have proposed extended rough set models to handle incomplete data while others proposed extensions of classical rough set models to deal with numerical data however there exist many data which contain numerical attributes and missing values simultaneously we propose in this paper an extended rough set model i e tolerance fuzzy rough set model to deal with this type of data characterized with numerical attributes and missing values that is incomplete numerical data discernibility matrices and discernibility functions for incomplete numerical information systems and incomplete numerical decision systems are defined to compute reducts or relative reducts meanwhile the relationship between the proposed tolerance fuzzy rough set model and the tolerance rough set model is also examined it is shown that the tolerance fuzzy rough set model is an extension of the tolerance rough set model finally uncertainty measurement is also investigated it is suggested that the proposed tolerance fuzzy rough set model provide an optional approach to incomplete numerical data

rough set approach to incomplete information systems abstract in the paper we present rough set approach to reasoning in incomplete information systems we propose reduction of knowledge that eliminates only that information which is not essential from the point of view of classification or decision making in our approach we make only one assumption about unknown values the real value of a missing attribute is one from the attribute domain however we do not assume which one we show how to find decision rules directly from such an incomplete decision table which are as little non deterministic as possible and have minimal number of conditions

on the nonnegative square root of the fourth derivative operators abstract the singularities of the nonnegative square root of the fourth derivative operator au u 4 are studied

efficient three dimensional delaunay triangulation with automatic point creation and imposed boundary constraints a method is described which constructs three dimensional unstructured tetrahedral meshes using the delaunay triangulation criterion several automatic point creation techniques will be highlighted and an algorithm will be presented which can ensure that given an initial surface triangulation which bounds a domain a valid boundary conforming assembly of tetrahedra will be produced statistics of measures of grid quality are presented for several grids the efficiency of the proposed procedure reduces the computer time for the generation of realistic unstructured tetrahedral grids to the order of minutes on workstations of modest computational capabilities

overlapping schwarz methods for unstructured spectral elements a parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied the spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron pyramid prism or cube in three dimensions the method is based on overlapping schwarz techniques applied to the schur complement of the discrete system and is implemented as a preconditioner for a krylov space method numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains the resulting elliptic solver can be used in navier stokes simulations using the spectral element code nektar

entropy measures and granularity measures for set valued information systems set valued information systems are generalized models of single valued information systems in this paper we propose two new relations for set valued information systems based on these two relations the concepts of knowledge information entropy knowledge rough entropy knowledge granulation and knowledge granularity measure are defined in set valued information systems and some properties are investigated moreover relationship between knowledge information entropy and knowledge granulation knowledge rough entropy and knowledge granularity measure are studied it is also shown that knowledge information entropy and knowledge granularity measure can be used to evaluate the certainty degree of knowledge in set valued information systems and knowledge rough entropy and knowledge granulation can be used to evaluate the uncertainty degree of knowledge in set valued information systems these results may supply a further understanding the essence of uncertainty and granularity in set valued information systems

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

the superiority of three way decisions in probabilistic rough set models three way decisions provide a means for trading off different types of classification error in order to obtain a minimum cost ternary classifier this paper compares probabilistic three way decisions probabilistic two way decisions and qualitative three way decisions of the standard rough set model it is shown that under certain conditions when considering the costs of different types of miss classifications probabilistic three way decisions are superior to the other two

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

approximations and uncertainty measures in incomplete information systems there are mainly two methodologies dealing with uncertainty measurement issue in rough set theory pure rough set approach and information theory approach pure rough set approach is based on the concepts of accuracy roughness and approximation accuracy proposed by pawlak information theory approach is based on shannon s entropy or its variants several authors have extended the information theory approach into incomplete information systems however there are few studies on extending the pure rough set approach to incomplete information systems this paper focuses on constructing uncertainty measures in incomplete information systems by pure rough set approach three types of definitions of lower and upper approximations and corresponding uncertainty measurement concepts including accuracy roughness and approximation accuracy are investigated theoretical analysis indicates that two of the three types can be used to evaluate the uncertainty in incomplete information systems experiments on incomplete real life data sets have been conducted to test the two selected types the first type and the third type of uncertainty measures results show that the two types of uncertainty measures are effective

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

a triangular spectral element method for elliptic and stokes problems abstract in this paper we study a triangular spectral element method based on a one to one mapping between the rectangle and the triangle we construct a new approximation space where the integral singularity brought by the mapping can be removed in a naive and stable way we build aquasi interpolation triangular spectral element approximation and analyze its approximation error based on this quasi interpolation spectral element approximation we put forward a new triangular spectral element method for the elliptic problems we present the approximation scheme analyze the convergence and do some experiments to test the effectiveness at last we implement this triangular spectral element method to solve the steady stokes problem

characterization of domains of fractional powers of certain operators arising in elastic systems and applications with x a hilbert space where a the elastic operator and b the damping operator are closed operators in x with dense domains q a and 9 b respectively moreover a is throughout assumed to be self adjoint and strictly positive stimulated by two conjectures raised in crl the most general of which referred to the case lx f described below we have already studied the operators and sb defined respectively by