on quick attribute reduction in decision theoretic rough set models compared with the pawlak rough set model decision theoretic rough set dtrs models have relatively powerful data processing capability for complex and low quality data however the non monotonicity of the criteria used in dtrs models causes many problems particularly the inefficiency of attribute reduction algorithms which greatly narrows their applications few systematic studies have been reported on how to construct efficient reduction methods for dtrs models this paper discusses this problem and constructs an efficient reduction algorithm for dtrs models incorporating three aspects taking advantage of common characteristics of data sets and employing division techniques a fast division approach to computing equivalence classes is proposed which is the most frequently used and the most time consuming basic operation in the reduction algorithm extracting a monotonic ingredient from decision systems an effective heuristic function is constructed for the reduction algorithm which can guide searches for the algorithm and has better ability to find shorter super reducts a novel algorithm is proposed to search the power set space of a given super reduct which has a shorter average search length than the existing backtracking methods finally the reduction algorithm is proposed that combines these improvements experimental results demonstrate that these improvements are effective and the proposed reduction algorithm is relatively efficient

block boundary value methods for solving volterra integral and integro differential equations reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind volterra integral equations and volterra integro differential equations these extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for volterra type equations numerical experiments are presented and the efficiency accuracy and stability of the schemes are confirmed

a class of compact boundary value methods applied to semi linear reaction diffusion equations abstract this paper deals with a class of compact boundary value methods cbvms for solving semi linear reaction diffusion equations slres the presented cbvms are constructed by combining a fourth order compact difference method cdm with the p order boundary value methods bvms where the former is for the spatial discretization and the latter for temporal discretization it is proven under some suitable conditions that the cbvms are locally stable and uniquely solvable and have fourth order accuracy in space and p order accuracy in time the computational effectiveness and accuracy of cbvms are further testified by applying the methods to the fisher equation besides these research we also extend the cbvms to solve the two component coupled system of slres the numerical experiment shows that the extended cbvms are effective and can arrive at the high precision

extensions and intentions in the rough set theory abstract the approach to rough set theory proposed in this paper is based on the mutual correspondence of the concepts of extension and intension it is different from the well known approaches in the literature in that the upper approximations and the lower approximations of unknown sets are considered as certain families of known sets this approach makes it possible to formulate necessary and sufficient conditions for the existence of operations on rough sets which are analogous to classical operations on sets the basic results presented in this paper based on certain ideas of the second author were formulated by the first author in his doctoral dissertation prepared under the supervision of the third author

boundary value methods for volterra integral and integro differential equations abstract new and effective quadrature rules generated by boundary value methods are introduced we employ the introduced quadrature rules to construct quadrature methods for the second kind volterra integral equations and volterra integro differential equations these methods are shown to be effective and possess excellent convergence properties the nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme numerical simulations are presented and confirm the efficiency and accuracy of the methods

block boundary value methods for the solution of ordinary differential equations block boundary value methods are considered and applied to solve initial value problems results on their convergence and stability properties are presented and their intermediate position between the class of multistep and runge kutta methods is established

on the relations between b2v ms and runge kutta collocation methods the principal aim of this paper is a rigorous analysis of the relations between block boundary value methods b 2v ms with minimal blocksize defined over a suitable nonuniform finer mesh and well known runge kutta collocation methods moreover a further aspect that will be briefly investigated is the construction of an extended finer mesh for building b 2v ms with nonminimal blocksize some advantages that may arise from the use of the so obtained methods will be also discussed

double quantitative decision theoretic rough set the probabilistic rough set prs and the graded rough set grs are two quantification models that measure relative and absolute quantitative information between the equivalence class and a basic concept respectively as a special prs model the decision theoretic rough set dtrs mainly utilizes the conditional probability to express relative quantification however it ignores absolute quantitative information of the overlap between equivalence class and the basic set and it cannot reflect the distinctive degrees of information and extremely narrow their applications in real life in order to overcome these defects this paper proposes a framework of double quantitative decision theoretic rough set dq dtrs based on bayesian decision procedure and grs two kinds of dq dtrs model are constructed which essentially indicate the relative and absolute quantification after further studies to discuss decision rules and the inner relationship between these two models we introduce an illustrative case study about the medical diagnosis to interpret and express the theories which is valuable for applying these theories to deal with practical issues

rough approximations on a complete completely distributive lattice with applications to generalized rough sets the generalizations of rough sets considered with respect to similarity relation covers and fuzzy relations are main research topics of rough set theory however these generalizations have shown less connection among each other and have not been brought into a unified framework which has limited the in depth research and application of rough set theory in this paper the complete completely distributive ccd lattice is selected as the mathematical foundation on which definitions of lower and upper approximations that form the basic concepts of rough set theory are proposed these definitions result from the concept of cover introduced on a ccd lattice and improve the approximations of the existing crisp generalizations of rough sets with respect to similarity relation and covers when t similarity relation is considered the existing fuzzy rough sets are the special cases of our proposed approximations on a ccd lattice thus these generalizations of rough sets are brought into a unified framework and a wider mathematical foundation for rough set theory is established

numerical approximation to a class of nonlinear hybrid system with distributed delay via block boundary value methods abstract this paper deals with the numerical approximation to a class of nonlinear hybrid system with distributed delay hsdd by combining block boundary value methods bbvms and reducible quadrature rules based on the underlying boundary value methods bvms a kind of extended bbvms are proposed for solving nonlinear hsdd under some suitable conditions an extended bbvm is proved to be uniquely solvable globally stable and convergent of order p where p is consistency order of the used underlying bvm the presented numerical experiments further illustrate the obtained theoretical results and computational effectiveness of the extended bbvms

convergence and stability of boundary value methods for ordinary differential equations a usual way to approximate the solution of initial value problems for ordinary differential equations is the use of a linear multistep formula if the formula has k steps k values are needed to obtain the discrete solution the continuous problem provides only the initial value it is customary to impose the additional k 1 conditions at the successive k 1 initial points however the class of methods obtained in this way suffers from heavy limitations summarized by the two dahlquist barriers it is also possible to impose the additional conditions at different grid points for example some conditions can be imposed at the initial points and the remaining ones at the final points the obtained methods called boundary value methods bvms do not have barriers whatsoever in this paper the question of convergence of bvms is discussed along with the linear stability theory some numerical examples on stiff test problems are also presented

rough sets and intelligent data analysis rough set based data analysis starts from a data table called an information system the information system contains data about objects of interest characterized in terms of some attributes often we distinguish in the information system condition and decision attributes such information system is called a decision table the decision table describes decisions in terms of conditions that must be satisfied in order to carry out the decision specified in the decision table with every decision table a set of decision rules called a decision algorithm can be associated it is shown that every decision algorithm reveals some well known probabilistic properties in particular it satisfies the total probability theorem and the bayes theorem these properties give a new method of drawing conclusions from data without referring to prior and posterior probabilities inherently associated with bayesian reasoning

order of convergence of one step methods for volterra integral equations of the second kind nice proofs of convergence and asymptotic expansions are known for one step methods for ordinary differential equations it is shown that these proofs can be generalized in a natural way to extended one step methods for volterra integral equations of the second kind furthermore the convergence of mixed one step methods is investigated for both types general volterra runge kutta methods are considered as examples

on the generalization of fuzzy rough sets rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets the existing generalizations of fuzzy rough sets are all based on special fuzzy relations fuzzy similarity relations t similarity relations it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches in this paper from the viewpoint of constructive approach we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined in axiomatic approach we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms the lattice and topological structures of fuzzy rough sets are also proposed in order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets a special lower approximation operator is applied to a fuzzy reasoning system which coincides with the mamdani algorithm

convergence and stability of multistep methods solving nonlinear initial value problems convergence and stability of initial and boundary value multistep methods are analyzed for a class of nonlinear problems satisfying a one sided lipschitz condition the linear multistep methods are recast to handle the numerical solution globally on the time interval this allows us to use the theory of toeplitz matrices to show that under suitable assumptions global properties of the exact solution are preserved by its numerical approximation in particular a new concept of stability which avoids the unpleasant passage to one leg methods is introduced

block boundary value methods for linear hamiltonian systems the problem of characterizing multistep methods suitable to efficiently approximate the solutions of linear hamiltonian systems is discussed showing that the appropriate methods should belong to the class of discrete boundary value methods bvms three families of such methods are proposed the presented methods have infinite regions of absolute stability and can be of any order in fact for every odd k there are k step methods of order up to 2k which is the maximum order reachable by a k step formula

exact solutions for functionally graded pressure vessels in a uniform magnetic field abstract analytical studies for magnetoelastic behavior of functionally graded material fgm cylindrical and spherical vessels placed in a uniform magnetic field subjected to internal pressure are presented exact solutions for displacement stress and perturbation of magnetic field vector in fgm cylindrical and spherical vessels are determined by using the infinitesimal theory of magnetoelasticity the material stiffness and magnetic permeability obeying a simple power law are assumed to vary through the wall thickness and poisson s ratio is assumed constant stresses and perturbation of magnetic field vector distributions depending on an inhomogeneous constant are compared with those of the homogeneous case and presented in the form of graphs the inhomogeneous constant which includes continuously varying volume fraction of the constituents is empirically determined the values used in this study are arbitrary chosen to demonstrate the effect of inhomogeneity on stresses and perturbation of magnetic field vector distributions

general linear methods for volterra integro differential equations with memory a new class of numerical methods for volterra integro differential equations with memory is developed the methods are based on the combination of general linear methods with compound quadrature rules sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established numerical examples illustrate the convergence and effectiveness of the numerical methods

nonlinear dynamic response and active vibration control for piezoelectric functionally graded plate the nonlinear dynamic response and active vibration control of the piezoelectric functionally graded plate are analyzed in this paper based on higher order shear plate theory and elastic piezoelectric theory the nonlinear geometric and constitutive relations of the piezoelectric functionally graded plate are established and then the nonlinear motion equations of the piezoelectric functionally graded plate are obtained through hamilton s variational principle the nonlinear active vibration control of the structure is carried out with adoption of the negative velocity feedback control algorithm by applying finite difference method the whole problem is solved by using iterative method synthetically in numerical examples the effects of mechanical load electric load the volume fraction and the geometric parameters on the dynamic response and vibration control of the piezoelectric fgm plate are investigated

the model of fuzzy variable precision rough sets the fuzzy rough set frs model has been introduced to handle databases with real values however frs was sensitive to misclassification and perturbation here misclassification means error or missing values in classification and perturbation means small changes of numerical data the variable precision rough sets vprss model was introduced to handle databases with misclassification however it could not effectively handle the real valued datasets now it is valuable from theoretical and practical aspects to combine frs and vprs so that a powerful tool which not only can handle numerical data but also is less sensitive to misclassification and perturbation can be developed in this paper we set up a model named fuzzy vprss fvprss by combining frs and vprs with the goal of making frs a special case first we study the knowledge representation ways of frs and vprs and then propose the set approximation operators of fvprs second we employ the discernibility matrix approach to investigate the structure of attribute reductions in fvprs and develop an algorithm to find all reductions third in order to overcome the np complete problem of finding all reductions we develop some fast heuristic algorithms to obtain one near optimal attribute reduction finally we compare fvprs with rs frs and several flexible rs based approaches with respect to misclassification and perturbation the experimental comparisons show the feasibility and effectiveness of fvprs

dynamic responses of piezoelectric hollow cylinders in an axial magnetic field abstract this paper presents an analytical solution for the interaction of electric potentials electric displacements elastic deformations and mechanical loads and describes electromagnetoelastic responses and perturbation of the magnetic field vector in a piezoelectric hollow cylinder subjected to sudden mechanical load and electric potential an interpolation method is applied to solve the volterra integral equation of the second kind caused by interactions between different physical fields by means of finite integral transforms laplace transforms and their inverse transforms the exact expressions for the dynamic responses of stresses electric displacements electric potentials and perturbation response of an axial magnetic field vector in the piezoelectric hollow cylinders are obtained the present method is suitable for piezoelectric hollow cylinders in an axial magnetic field subjected to arbitrary mechanical loads and electrical potential shocks finally numerical results are carried out and discussed

a novel method for attribute reduction of covering decision systems attribute reduction has become an important step in pattern recognition and machine learning tasks covering rough sets as a generalization of classical rough sets have attracted wide attention in both theory and application this paper provides a novel method for attribute reduction based on covering rough sets we review the concepts of consistent and inconsistent covering decision systems and their reducts and we develop a judgment theorem and a discernibility matrix for each type of covering decision system furthermore we present some basic structural properties of attribute reduction with covering rough sets based on a discernibility matrix we develop a heuristic algorithm to find a subset of attributes that approximate a minimal reduct finally the experimental results for uci data sets show that the proposed reduction approach is an effective technique for addressing numerical and categorical data and is more efficient than the method presented in the paper d g chen c z wang q h hu a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets information sciences 177 17 2007 3500 3518

hybrid attribute reduction based on a novel fuzzy rough model and information granulation feature subset selection has become an important challenge in areas of pattern recognition machine learning and data mining as different semantics are hidden in numerical and categorical features there are two strategies for selecting hybrid attributes discretizing numerical variables or numericalize categorical features in this paper we introduce a simple and efficient hybrid attribute reduction algorithm based on a generalized fuzzy rough model a theoretic framework of fuzzy rough model based on fuzzy relations is presented which underlies a foundation for algorithm construction we derive several attribute significance measures based on the proposed fuzzy rough model and construct a forward greedy algorithm for hybrid attribute reduction the experiments show that the technique of variable precision fuzzy inclusion in computing decision positive region can get the optimal classification performance number of the selected features is the least but accuracy is the best

a new effect based roughness measure for attribute reduction in information system roughness measure a quantitative index for processing uncertain information using fuzzy set theory is the basis of resource management system optimization and many other decision making problems the construction of a roughness measure which reflects different decision preferences has important theoretical and practical value in this paper we first analyze the characteristics and shortcomings of the existing methods for measuring roughness following this a new effect based roughness measure model is established using a combination of a basic measure factor the lower or upper accuracy of rough sets entitled the effective rough degree erd next the characteristics of erd are analyzed in combination with different synthesis functions and several further necessary and sufficient conditions are given finally we propose an erd based attribute reduction method abbreviated as erd rm and then discuss the differences and relationships between erd rm and existing reduction methods the theoretical analysis and practical applications show that erd has good structural features and interpretability and can integrate decision preference into the measure system in a straightforward manner

stability analysis of volterra delay integro differential equations and their backward differentiation time discretization we study the stability of the analytical solutions of initial value problems of a general class of systems of volterra delay integro differential equations numerical methods based on backward differentiation formulae and repeated quadrature formulae are suggested nonlinear and linear stability conditions for the presented methods are derived

related family a new method for attribute reduction of covering information systems in terms of attribute reduction of covering based rough sets the discernibility matrix is used as a conventional method to compute all attribute reducts however it is inapplicable to attribute reduction in certain circumstances in this article a new method referred to as the related family is introduced to compute all attribute reducts and relative attribute reducts for covering rough sets its core idea is to remove superfluous attributes while keeping the approximation space of covering information system unchanged the related family method is more powerful than the discernibility matrix method since the former can handle complicated cases that could not be handled by the latter in addition a simplified version of the related family and its corresponding heuristic algorithm are also presented

a new approach to fuzzy rule generation fuzzy extension matrix this paper proposes a new approach to fuzzy rule generation from a set of examples with fuzzy representation the new approach called fuzzy extension matrix incorporates the fuzzy entropy to search for paths and generalizes the concept of crisp extension matrix by discussing paths of the fuzzy extension matrix a new heuristic algorithm for generating fuzzy rules is introduced compared with the crisp extension matrix the proposed method has the capability of handling fuzzy representation and tolerating noisy data or missing data a case study shows that the proposed heuristic algorithm partially inherits the advantages from the crisp case such as simplicity of rules and high learning accuracy the proposed approach offers a new practical way to automatically acquire imprecise knowledge

a global convergence theorem for a class of parallel continuous explicit runge kutta methods and vanishing lag delay differential equations iterated continuous extensions ices are continuous explicit runge kutta methods developed for the numerical solution of evolutionary problems in ordinary and delay differential equations ddes ices have a particular role in the explicit solution of ddes with vanishing lags they may be regarded as parallel continuous explicit runge kutta pcerk methods as they allow one to take advantage of parallel architectures ices can also be related to a collocation method the purpose of this paper is to provide a theorem giving the global order of convergence for variable step implementations of ices applied to state dependent ddes with and without vanishing lags implications of the theory for the implementation of this class of methods are discussed and demonstrated the results establish that our approach allows the construction of pcerk methods whose order of convergence is restricted only by the continuity of the solution

relationships between generalized rough sets based on covering and reflexive neighborhood system rough set theory is an important tool to deal with uncertainty granularity and incompleteness of knowledge the covering based rough sets is one of the most important extensions of the classical pawlak rough sets in covering based rough set theory many types of rough sets were already defined in the literature to find out the relationships among different types of covering based rough sets and give an axiomatic definition is a very important issue the generalized rough sets in neighborhood system is another important extension of the classical pawlak rough sets in this paper we investigate relationships between covering based rough sets and generalized rough sets in neighborhood system then twenty three types of the element based the granule based and the subsystem based definitions of covering approximation operators are unified under the framework of generalized approximation operators in neighborhood systems moreover properties of the generalized approximation operators in neighborhood systems are presented

a new measure of uncertainty based on knowledge granulation for rough sets in rough set theory accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification although these measures are effective they have some limitations when the lower upper approximation of a set under one knowledge is equal to that under another knowledge to overcome these limitations we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table an axiomatic definition of knowledge granulation for an information system is given under which these three measures are modified theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table respectively and have a much simpler and more comprehensive form than the existing ones