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

an efficient numerical method for singular perturbation problems in this paper we propose a method for the numerical solution of singularly perturbed two point boundary value problems bvps first we develop two schemes to integrate initial value problem ivp for system of two first order differential equations and then by using these schemes we solve the bvp precisely we convert the second order bvp into a system of first order differential equations and then apply the numerical schemes to obtain the solution in order to get an initial condition for the system we use the asymptotic approximate solution error estimates are derived and numerical examples are provided to illustrate the present method

uniqueness and error estimates for solutions to higher order boundary value problems abstract little work has been done for the error estimates of the homotopy analysis method for general 2 n th order linear and nonlinear differential equations with lidstone boundary conditions we obtain sharp upper bounds for the absolute error of the approximations given by the homotopy analysis method to achieve this goal the existence and uniqueness of solutions to these differential equations are considered numerical results confirm the theoretical predictions

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

parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers this article presents a numerical method to solve singularly perturbed turning point problems exhibiting two exponential boundary layers classical finite difference schemes do not yield parameter uniform convergent results on a uniform mesh in general robust computational techniques for boundary layers chapman hall london crc press boca raton fl 2000 in order to overcome this difficulty we propose an appropriate piecewise uniform shishkin mesh and apply the classical finite difference schemes on this mesh error estimates are derived by decomposing the solution into smooth and singular components the present method is layer resolving as well as parameter uniform convergent numerical examples are presented to show the applicability and efficiency of the method

restricted p isometry properties of nonconvex matrix recovery recently a nonconvex relaxation of low rank matrix recovery lmr called the schatten p quasi norm minimization 0 p 1 was introduced instead of the previous nuclear norm minimization in order to approximate the problem of lmr closer in this paper we introduce a notion of the restricted p isometry constants 0 p 1 and derive a p rip condition for exact reconstruction of lmr via schatten p quasi norm minimization in particular we determine how many random gaussian measurements are needed for the p rip condition to hold with high probability which gives a theoretical result that it needs fewer measurements with small p for exact recovery via schatten p quasi norm minimization than when p 1

approximations and reducts with covering generalized rough sets the covering generalized rough sets are an improvement of traditional rough set model to deal with more complex practical problems which the traditional one cannot handle it is well known that any generalization of traditional rough set theory should first have practical applied background and two important theoretical issues must be addressed the first one is to present reasonable definitions of set approximations and the second one is to develop reasonable algorithms for attributes reduct the existing covering generalized rough sets however mainly pay attention to constructing approximation operators the ideas of constructing lower approximations are similar but the ideas of constructing upper approximations are different and they all seem to be unreasonable furthermore less effort has been put on the discussion of the applied background and the attributes reduct of covering generalized rough sets in this paper we concentrate our discussion on the above two issues we first discuss the applied background of covering generalized rough sets by proposing three kinds of datasets which the traditional rough sets cannot handle and improve the definition of upper approximation for covering generalized rough sets to make it more reasonable than the existing ones then we study the attributes reduct with covering generalized rough sets and present an algorithm by using discernibility matrix to compute all the attributes reducts with covering generalized rough sets with these discussions we can set up a basic foundation of the covering generalized rough set theory and broaden its applications

a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets traditional rough set theory is mainly used to extract rules from and reduce attributes in databases in which attributes are characterized by partitions while the covering rough set theory a generalization of traditional rough set theory does the same yet characterizes attributes by covers in this paper we propose a way to reduce the attributes of covering decision systems which are databases characterized by covers first we define consistent and inconsistent covering decision systems and their attribute reductions then we state the sufficient and the necessary conditions for reduction finally we use a discernibility matrix to design algorithms that compute all the reducts of consistent and inconsistent covering decision systems numerical tests on four public data sets show that the proposed attribute reductions of covering decision systems accomplish better classification performance than those of traditional rough sets

on the theorem of uniform recovery of random sampling matrices we consider two theorems from the theory of compressive sensing mainly a theorem concerning uniform recovery of random sampling matrices where the number of samples needed in order to recover an s sparse signal from linear measurements with high probability is known to be m s ln s 3 lnn we present new and improved constants together with what we consider to be a more explicit proof a proof that also allows for a slightly larger class of m n matrices by considering what is called effective sparsity we also present a condition on the so called restricted isometry constants s ensuring sparse recovery via l 1 minimization we show that is sufficient and that this can be improved further to almost allow for a sufficient condition of the type

relationship between generalized rough sets based on binary relation and covering rough set theory is a powerful tool for dealing with uncertainty granularity and incompleteness of knowledge in information systems this paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering based rough sets and the generalized rough sets based on binary relation firstly we present basic concepts and properties of this kind of rough sets then we investigate the relationships between this type of generalized rough sets and other five types of covering based rough sets the major contribution in this paper is that we establish the equivalency between this type of covering based rough sets and a type of binary relation based rough sets through existing results in binary relation based rough sets we present axiomatic systems for this type of covering based lower and upper approximation operations in addition we explore the relationships among several important concepts such as minimal description reduction representative covering exact covering and unary covering in covering based rough sets investigation of this type of covering based will benefit to our understanding of other types of rough sets based on covering and binary relation

an efficient accelerator for attribute reduction from incomplete data in rough set framework feature selection attribute reduction from large scale incomplete data is a challenging problem in areas such as pattern recognition machine learning and data mining in rough set theory feature selection from incomplete data aims to retain the discriminatory power of original features to address this issue many feature selection algorithms have been proposed however these algorithms are often computationally time consuming to overcome this shortcoming we introduce in this paper a theoretic framework based on rough set theory which is called positive approximation and can be used to accelerate a heuristic process for feature selection from incomplete data as an application of the proposed accelerator a general feature selection algorithm is designed by integrating the accelerator into a heuristic algorithm we obtain several modified representative heuristic feature selection algorithms in rough set theory experiments show that these modified algorithms outperform their original counterparts it is worth noting that the performance of the modified algorithms becomes more visible when dealing with larger data sets

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

axiomatization and conditions for neighborhoods in a covering to form a partition in this paper we study the axiomatic issue of a type of covering upper approximation operations this issue was proposed as an open problem we also further some known results by using only a single covering approximation operator to characterize the conditions for neighborhood n x x u to form a partition of universe u

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

numerical solution of 2d singularly perturbed reaction diffusion system with multiple scales abstract in this article a robust numerical method is studied to approximate singularly perturbed system of reaction diffusion problems with multiple scales the analytical properties of the exact solution have been studied the numerical method consists of the classical central difference scheme on a shishkin mesh for spatial semidiscretization processes and the implicit euler scheme on a uniform time stepping for temporal derivative the error estimate is deduced which exhibits that the numerical approximation is uniformly convergent of almost second order in spatial variable and first order in temporal variable numerical experiments are given which reveals the effectiveness of the proposed scheme

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

the further investigation of covering based rough sets uncertainty characterization similarity measure and generalized models the notion of rough sets was originally proposed by pawlak in pawlak s rough set theory the equivalence relation or partition plays an important role however the equivalence relation or partition is restrictive for many applications because it can only deal with complete information systems this limits the theory s application to a certain extent therefore covering based rough sets are derived by replacing the partitions of a universe with its coverings this paper focuses on the further investigation of covering based rough sets firstly we discuss the uncertainty of covering in the covering approximation space and show that it can be characterized by rough entropy and the granulation of covering secondly since it is necessary to measure the similarity between covering rough sets in practical applications such as pattern recognition image processing and fuzzy reasoning we present an approach which measures these similarities using a triangular norm we show that in a covering approximation space a triangular norm can induce an inclusion degree and that the similarity measure between covering rough sets can be given according to this triangular norm and inclusion degree thirdly two generalized covering based rough set models are proposed and we employ practical examples to illustrate their applications finally relationships between the proposed covering based rough set models and the existing rough set models are also made

richardson extrapolation for a singularly perturbed turning point problem with exponential boundary layers we consider an interior turning point problem with two exponential boundary layers its discretisation on a piecewise uniform shishkin mesh yields a scheme which is uniformly convergent measured in the discrete maximum norm of almost order one richardson extrapolation improves the accuracy to o n 2 ln 2 n both can be proved under the assumption e c n 1

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

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

a computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers singularly perturbed turning point problems tpps for second order ordinary differential equations des exhibiting twin boundary layers are considered in order to obtain numerical solution of these problems a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods in this method the given internal domain of definition of differential equation is divided into four subintervals and we solve the differential equation in each interval and combine the solution the implementation of the method on parallel architectures is mentioned error estimates are derived and numerical examples are given

does ell _ p minimization outperform ell _ 1 minimization in many application areas ranging from bioinformatics to imaging we are faced with the following question can we recover a sparse vector x o in mathbb r n from its undersampled set of noisy observations y in mathbb r n y a x o w the last decade has witnessed a surge of algorithms and theoretical results to address this question one of the most popular schemes is the ell p regularized least squares given by the following formulation hat x gamma p in arg min x 1 2 y ax 2 2 gamma x p p where p in 0 1 among these optimization problems the case p 1 also known as lasso is the best accepted in practice for the following two reasons first thanks to the extensive studies performed in the fields of high dimensional statistics and compressed sensing we have a clear picture of lasso s performance second it is convex and efficient algorithms exist for finding its global minima unfortunately neither of the above two properties hold for 0 leq p however they are still appealing because of the following folklores in the high dimensional statistics first hat x gamma p is closer to x o than hat x gamma 1 second if we employ iterative methods that aim to converge to a local minima of arg min x 1 2 y ax 2 2 gamma x p p then under good initialization these algorithms converge to a solution that is still closer to x o than hat x gamma 1 in spite of the existence of plenty of empirical results that support these folklore theorems the theoretical progress to establish them has been very limited this paper aims to study the above mentioned folklore theorems and establish their scope of validity starting with approximate message passing amp algorithm as a heuristic method for solving ell p regularized least squares we study the following questions first what is the impact of initialization on the performance of the algorithm second when does the algorithm recover the sparse signal x o under a good initialization third when does the algorithm converge to the sparse signal regardless of the initialization studying these questions will not only shed light on the second folklore theorem but also lead us to the answer the first one i e the performance of the global optima hat x gamma p for that purpose we employ the replica analysis 1 to show the connection between the solution of amp and hat x gamma p in the asymptotic settings this enables us to compare the accuracy of hat x gamma p and hat x gamma 1 in particular we will present an accurate characterization of the phase transition and noise sensitivity of ell p regularized least squares for every 0 leq p leq 1 our results in the noiseless setting confirm that ell p regularized least squares if gamma is tuned optimally exhibits the same phase transition for every 0 leq p and this phase transition is much better than that of lasso furthermore we show that in the noisy setting there is a major difference between the performance of ell p regularized least squares with different values of p for instance we will show that for very small and very large measurement noises p 0 and p 1 outperform the other values of p respectively 1 replica method is a widely accepted heuristic in statistical physics for analyzing large disordered systems

alternating direction numerical scheme for singularly perturbed 2d degenerate parabolic convection diffusion problems abstract in this article we study the numerical solution of singularly perturbed 2d degenerate parabolic convection diffusion problems on a rectangular domain the solution of this problem exhibits parabolic boundary layers along x 0 y 0 and a corner layer in the neighborhood of 0 0 first we use an alternating direction implicit finite difference scheme to discretize the time derivative of the continuous problem on a uniform mesh in the temporal direction then to discretize the spatial derivatives of the resulting time semidiscrete problems we apply the upwind finite difference scheme on a piecewise uniform shishkin mesh we derive error estimate for the proposed numerical scheme which shows that the scheme is e uniformly convergent of almost first order up to a logarithmic factor in space and first order in time some numerical results have been carried out to validate the theoretical results

reproducing kernel method for solving singularly perturbed differential difference equations with boundary layer behavior in hilbert space abstract we consider singularly perturbed differential difference equation which contains negative shift in the differentiated term with boundary layer behavior rkhsm reproducing kernel hilbert space method without gram schmidt orthogonalization process is considered in the present paper we decompose the domain of the problem into two subintervals one of them has not the boundary layer and the other one has the side of the interval in which the boundary layer exists is important if the boundary layer of this problem exists on the left side of interval the rkhsm will provide a proper approximation of solution otherwise for the implement of rkhsm we need to change the variable of the singularly perturbed problem to shift the boundary layer region

on the error of estimating the sparsest solution of underdetermined linear systems let a be an n m matrix with m n and suppose that the underdetermined linear system as x admits a sparse solution s0 for which s0 0 1 2 spark a such a sparse solution is unique due to a well known uniqueness theorem suppose now that we have somehow a solution s as an estimation of s0 and suppose that s is only approximately sparse that is many of its components are very small and nearly zero but not mathematically equal to zero is such a solution necessarily close to the true sparsest solution more generally is it possible to construct an upper bound on the estimation error s s0 2 without knowing s0 the answer is positive and in this paper we construct such a bound based on minimal singular values of submatrices of a we will also state a tight bound which is more complicated but besides being tight enables us to study the case of random dictionaries and obtain probabilistic upper bounds we will also study the noisy case that is where x as n moreover we will see that where s0 0 grows to obtain a predetermined guaranty on the maximum of s s0 2 s is needed to be sparse with a better approximation this can be seen as an explanation to the fact that the estimation quality of sparse recovery algorithms degrades where s0 0 grows

a numerical method for singularly perturbed turning point problems with an interior layer the objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer the method is based on the asymptotic expansion technique and the reproducing kernel method rkm the original problem is reduced to interior layer and regular domain problems the regular domain problems are solved by using the asymptotic expansion method the interior layer problem is treated by the method of stretching variable and the rkm four numerical examples are provided to illustrate the effectiveness of the present method the results of numerical examples show that the present method can provide very accurate approximate solutions

a computational method for solving singular perturbation problems abstract a class of singularly perturbed two point boundary value problems for second order ordinary differential equations is considered in order to solve them a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods the proposed method is distinguished by the following facts first we divide the given interval the domain of definition of differential equation into two subintervals called inner and outer regions then we solve the differential equation over both the regions as two point boundary value problems the terminal boundary condition of the inner region is obtained using the zero order asymptotic expansion of the solution some numerical examples are given to illustrate the method

multiple description l_ 1 compression multiple descriptions mds is a method to obtain reliable signal transmissions on erasure channels an md encoder forms several descriptions of the signal and each description is independently transmitted across an erasure channel the reconstruction quality then depends on the set of received descriptions in this paper we consider the design of redundant descriptions in an md setup using l1 minimization with euclidean distortion constraints in this way we are able to obtain sparse descriptions using convex optimization the proposed method allows for an arbitrary number of descriptions and supports both symmetric and asymmetric distortion design we show that mds with partial overlapping information corresponds to enforcing coupled constraints in the proposed convex optimization problem to handle the coupled constraints we apply dual decompositions which makes first order methods applicable and thereby admit solutions for large scale problems e g coding entire images or image sequences we show by examples that the proposed framework generates non trivial sparse descriptions and non trivial refinements we finally show that the sparse descriptions can be quantized and encoded using off the shell encoders such as the set partitioning in hierarchical trees spiht encoder however the proposed method shows a rate distortion loss compared to state of the art image md encoders

a unified framework for characterizing rough sets with evidence theory in various approximation spaces abstract different rough set models may be numerically characterized with evidence theory by adopting different methodologies that is to say the evidence theory based characteristics of rough sets vary with different approximation spaces and different rough set approximations to address this issue this paper proposes a theoretic framework within which rough set theory can be unifiedly characterized by evidence theory no matter what types of rough set approximations and approximation spaces they are the main works are presented as follows first we declare a principle that in any given approximation space a belief structure can be constructed based on which the lower and upper approximation operators can be measured by the belief and plausibility functions if and only if a basic condition is satisfied second in a given decision approximation space a belief structure is also derived based on which the lower and upper approximation operators can be always measured by the belief and plausibility functions without any additional condition as theoretical applications we employ the new principles to examine the evidence theory based characteristics of four types of substantial rough sets including the covering rough sets in covering approximation spaces the decision theoretic rough sets in pawlak approximation spaces the pawlak rough sets in pawlak decision approximation spaces and the multigranulation rough sets in pawlak decision approximation spaces respectively the proposed framework is applicable to various approximation spaces and various rough set approximations and hence can make us clear the basic principles for relating rough set theory with evidence theory

topological approaches to covering rough sets rough sets a tool for data mining deal with the vagueness and granularity in information systems this paper studies covering based rough sets from the topological view we explore the topological properties of this type of rough sets study the interdependency between the lower and the upper approximation operations and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation lastly axiomatic systems for the lower approximation operation and the upper approximation operation are constructed

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

knowledge reduction of dynamic covering decision information systems when varying covering cardinalities in covering based rough set theory non incremental approaches are time consuming for performing knowledge reduction of dynamic covering decision information systems when the cardinalities of coverings change as a result of object immigration and emigration because computing approximations of sets is an important step for knowledge reduction of dynamic covering decision information systems efficient approaches to calculating the second and sixth lower and upper approximations of sets using the type 1 and type 2 characteristic matrices respectively are essential in this paper we provide incremental approaches to computing the type 1 and type 2 characteristic matrices of dynamic coverings whose cardinalities vary with the immigration and emigration of objects we also design incremental algorithms to compute the second and sixth lower and upper set approximations experimental results demonstrate that the incremental approaches effectively improve the efficiency of set approximation computation finally we employ several examples to illustrate the feasibility of the incremental approaches for knowledge reduction of dynamic covering decision information systems when increasing the cardinalities of coverings

majorization minimization algorithms for wavelet based image restoration standard formulations of image signal deconvolution under wavelet based priors regularizers lead to very high dimensional optimization problems involving the following difficulties the non gaussian heavy tailed wavelet priors lead to objective functions which are nonquadratic usually nondifferentiable and sometimes even nonconvex the presence of the convolution operator destroys the separability which underlies the simplicity of wavelet based denoising this paper presents a unified view of several recently proposed algorithms for handling this class of optimization problems placing them in a common majorization minimization mm framework one of the classes of algorithms considered when using quadratic bounds on nondifferentiable log priors shares the infamous singularity issue si of iteratively re weighted least squares irls algorithms the possibility of having to handle infinite weights which may cause both numerical and convergence issues in this paper we prove several new results which strongly support the claim that the si does not compromise the usefulness of this class of algorithms exploiting the unified mm perspective we introduce a new algorithm resulting from using bounds for nonconvex regularizers the experiments confirm the superior performance of this method when compared to the one based on quadratic majorization finally an experimental comparison of the several algorithms reveals their relative merits for different standard types of scenarios

quintic b spline method for solving second order linear and nonlinear singularly perturbed two point boundary value problems in this paper we have studied a numerical scheme to solve second order singularly perturbed two point linear and nonlinear boundary value problems the boundary layer of this type of problems exhibits at one end left or right point of the domain due to the presence of perturbation parameter the quintic b spline method and suitable piecewise uniform shishkin mesh have been used linear and nonlinear second order singularly perturbed boundary value problems have been solved by the present method the convergence analysis is also provided and the method is shown to have uniform convergence of fourth order numerical results have demonstrated the efficiency of the present method

reduction and axiomization of covering generalized rough sets this paper investigates some basic properties of covering generalized rough sets and their comparison with the corresponding ones of pawlak s rough sets a tool for data mining the focus here is on the concepts and conditions for two coverings to generate the same covering lower approximation or the same covering upper approximation the concept of reducts of coverings is introduced and the procedure to find a reduct for a covering is given it has been proved that the reduct of a covering is the minimal covering that generates the same covering lower approximation or the same covering upper approximation so this concept is also a technique to get rid of redundancy in data mining furthermore it has been shown that covering lower and upper approximations determine each other finally a set of axioms is constructed to characterize the covering lower approximation operation

information theoretic measures associated with rough set approximations in rough set theory some information theoretic measures of uncertainty and granularity have been proposed a common feature of these measures is that they are only dependent on the partitions and the cardinality of a universe which means that they are independent of the lower and upper approximations of rough sets this seems somewhat unreasonable since the basic idea of rough set theory is to describe incomplete or inexact concepts by the lower and upper approximations in light of this we develop a new pair of information theoretic entropy and co entropy functions associated to partitions and approximations in this paper such functions are used to measure the uncertainty and granularity of an approximation space after introducing the novel notions of entropy and co entropy we then examine their properties in particular we disclose the relationship of co entropies between different universes the theoretical development is accompanied by illustrative numerical examples

rough matroids based on relations rough sets provide an efficient tool for attribute reduction and rule extraction however many important problems in rough set theory including attribute reduction are np hard and therefore the algorithms for solving them are usually greedy as a generalization of linear independence in vector spaces matroids have wide applications in diverse fields particularly in greedy algorithm design in this paper we propose an integration of rough sets and matroids to exploit the advantages of both theories specifically we present definitions of lower and upper rough matroids based on relations from the viewpoint of approximation operators it is interesting that lower rough matroids based on partial orders coincide with poset matroids which are a well known generalization of matroids a matroid is represented by the lower rough matroid based on both a partial order and an equivalence relation lower and upper rough matroids based on equivalence relations coincide with each other finally we present some special types of examples of rough matroids from the viewpoints of approximations and graphs these interesting results demonstrate the potential for the combination between rough sets and matroids

a higher order numerical scheme for singularly perturbed parabolic turning point problems exhibiting twin boundary layers abstract in this article a parameter uniform numerical method is presented to solve one dimensional singularly perturbed parabolic convection diffusion multiple turning point problems exhibiting two exponential boundary layers we study the asymptotic behaviour of the solution and its partial derivatives the problem is discretized using the implicit euler method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized shishkin mesh the scheme is shown to be e uniformly convergent of order one in time direction and order two in spatial direction upto a logarithmic factor numerical experiments are conducted to validate the theoretical results comparison is done with the upwind scheme on a uniform mesh as well as on the standard shishkin mesh to demonstrate the higher order accuracy of the proposed scheme on a generalized shishkin mesh

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

an entropy based uncertainty measurement approach in neighborhood systems abstract uncertainty measures can provide us with principled methodologies to analyze uncertain data and unveil the substantive characteristics of the data sets accuracy and roughness proposed by pawlak are two main tools to deal with uncertainty measurement issue in rough set theory many uncertainty measure methodologies for discrete valued information systems or discrete valued decision systems have been developed however there are only limited on the uncertainty measurement for neighborhood systems in this paper we address the issues of uncertainty of a neighborhood system and extend the traditional accuracy and roughness measures to deal with neighborhood systems in particular a concept called neighborhood entropy is first introduced to evaluate the uncertainty of a neighborhood information system consequently the entropy based roughness and approximation roughness measures of neighborhood system are presented theoretical analysis indicates that entropy based measures can be used to evaluate the uncertainty in neighborhood systems experiments are conducted on artificial data sets and standard uci data sets to test our proposed methodologies results demonstrate that the entropy based measures are effective and valid for evaluating the uncertainty of neighborhood systems

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

an adaptive moving grid method for a system of singularly perturbed initial value problems a system of first order singularly perturbed initial value problems is considered the system is discretized by a backward euler difference scheme for which a priori error analysis in the maximum norm is constructed it is shown from the a priori error bound that there exists a mesh with n subintervals that gives optimal error bound of o n 1 which is robust with respect to the perturbation parameters a partly heuristic argument based on a priori error analysis leads to a suitable monitor function based on an a posteriori error bound a first order rate of convergence independent of all perturbation parameters is established a linear and a nonlinear examples are tested and the numerical results are provided to demonstrate the effectiveness of our adaptive moving grid method

a numerical algorithm for singular perturbation problems exhibiting weak boundary layers abstract a class of singularly perturbed two point boundary value problems bvp for second order ordinary differential equations is considered here to avoid the numerical difficulties in the solution to these problems we divide the domain into two subdomains the first bvp is a layer domain problem and the second bvp is a regular domain problem error estimates are derived for the numerical solution numerical examples are provided in support of the proposed method

knowledge reduction in random information systems via dempster shafer theory of evidence knowledge reduction is one of the main problems in the study of rough set theory this paper deals with knowledge reduction in random information systems based on dempster shafer theory of evidence the concepts of belief and plausibility reducts in random information systems are first introduced it is proved that both of belief reduct and plausibility reduct are equivalent to classical reduct in random information systems the relative belief and plausibility reducts in consistent and inconsistent random decision systems are then proposed and compared to the relative reduct and relationships between the new reducts and some existing ones are examined

relationship among basic concepts in covering based rough sets uncertainty and incompleteness of knowledge are widespread phenomena in information systems rough set theory is a tool for dealing with granularity and vagueness in data analysis rough set method has already been applied to various fields such as process control economics medical diagnosis biochemistry environmental science biology chemistry psychology and conflict analysis covering based rough set theory is an extension to classical rough sets in covering based rough sets there exist several basic concepts such as reducible elements of a covering minimal descriptions unary coverings and the property that the intersection of any two elements is the union of finite elements in this covering these concepts appeared in the literature of covering based rough sets separately in this paper we study the relationships between them in particular we establish the equivalence of the unary covering and the covering with the property that the intersection of any two elements is the union of finite elements in this covering we also investigate the relationship between the covering lower approximation operation and the interior operator a characterization of the interior operator by the covering lower approximation operation is presented in this paper correspondingly we study the relationship between the covering upper approximation operation and the closure operator in addition we explore the conditions under which the covering upper approximation operation is monotone the study of the relationships between these concepts will help us have a better understanding of covering based rough sets

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

boundary value problems occurring in plate deflection theory abstract the paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods

a proof of conjecture on restricted isometry property constants delta _ tk left 0 t frac 4 3 right in this paper we give a complete answer to the conjecture on restricted isometry property rip constants delta tk 0 which was proposed by t cai and a zhang we have shown that when 0 the condition delta tk is sufficient to guarantee the exact recovery for all k sparse signals in the noiseless case via the constrained ell 1 norm minimization these bounds are sharp in the sense that for any epsilon 0 delta tk cannot guarantee the exact recovery of some k sparse signals furthermore it will be shown that similar characterizations also hold for low rank matrix recovery thus combined with t cai and a zhang s work a complete characterization for sharp rip constants delta tk for all t 0 is obtained to guarantee the exact recovery of all k sparse signals and matrices with rank at most k by ell 1 norm minimization and nuclear norm minimization respectively noisy cases and approximately sparse cases are also considered to solve the conjecture we construct a few identities so that rip of order tk which is the target of our main results can be perfectly applied to them

covering based rough set approximations we propose a framework for the study of covering based rough set approximations three equivalent formulations of the classical rough sets are examined by using equivalence relations partitions and s algebras respectively they suggest the element based the granule based and the subsystem based definitions of approximation operators covering based rough sets are systematically investigated by generalizing these formulations and definitions a covering of universe of objects is used to generate different neighborhood operators neighborhood systems coverings and subsystems of the power set of the universe they are in turn used to define different types of generalized approximation operators within the proposed framework we review and discuss covering based approximation operators according to the element granule and subsystem based definitions

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

restricted isometry constants where ell p sparse recovery can fail for 0 p leq 1 this paper investigates conditions under which the solution of an underdetermined linear system with minimal lscrp norm 0 p les 1 is guaranteed to be also the sparsest one matrices are constructed with restricted isometry constants ric delta2m arbitrarily close to 1 radic2 ap 0 707 where sparse recovery with p 1 fails for at least one m sparse vector as well as matrices with delta2m arbitrarily close to one where lscr1 minimization succeeds for any m sparse vector this highlights the pessimism of sparse recovery prediction based on the ric and indicates that there is limited room for improving over the best known positive results of foucart and lai which guarantee that lscr1 minimization recovers all m sparse vectors for any matrix with delta2m 2 3 radic2 7 ap 0 4531 these constructions are a by product of tight conditions for lscrp recovery 0 les p les 1 with matrices of unit spectral norm which are expressed in terms of the minimal singular values of 2m column submatrices compared to lscr1 minimization lscrp minimization recovery failure is shown to be only slightly delayed in terms of the ric values furthermore in this case the minimization is nonconvex and it is important to consider the specific minimization algorithm being used it is shown that when lscrp optimization is attempted using an iterative reweighted lscr1 scheme failure can still occur for delta2m arbitrarily close to 1 radic2

on measurements of covering rough sets based on granules and evidence theory covering rough sets generalize classical rough sets by relaxing partitions of the universe to coverings the existing research of covering rough sets mainly concentrates on comparing of different kinds of approximation operators and designing algorithms for attribute reductions of decision systems less efforts have been made in measurements of covering rough sets to enrich this field in this paper we investigate the measurements of covering rough sets based on granules and evidence theory which can be used to solve more complex practical problems firstly it is shown that the most common sixteen pairs of covering approximation operators can be grouped into five pairs of dual covering approximation operators and six pairs of non dual covering approximation operators according to definitions based on element granule and subsystem secondly we define the rough degree and precision for the purpose of characterizing the uncertainty of these summarized covering approximation operators under a unified framework the underlying uncertainty of covering rough sets results from the roughness of granulation of coverings finally we investigate the relationships between the covering approximation operators and evidence theory it shows that some kinds of covering approximation operators can be characterized by belief and plausibility functions while others do not share this property the sufficient conditions under which covering approximation operators can be measured by belief and plausibility functions are further explored and the corresponding counterexamples are given to illustrate those that cannot be measured based on above discussions we develop a basic framework of numerical characterization on measurements of covering rough sets with granules and evidence theory

positive approximation an accelerator for attribute reduction in rough set theory feature selection is a challenging problem in areas such as pattern recognition machine learning and data mining considering a consistency measure introduced in rough set theory the problem of feature selection also called attribute reduction aims to retain the discriminatory power of original features many heuristic attribute reduction algorithms have been proposed however quite often these methods are computationally time consuming to overcome this shortcoming we introduce a theoretic framework based on rough set theory called positive approximation which can be used to accelerate a heuristic process of attribute reduction based on the proposed accelerator a general attribute reduction algorithm is designed through the use of the accelerator several representative heuristic attribute reduction algorithms in rough set theory have been enhanced note that each of the modified algorithms can choose the same attribute reduct as its original version and hence possesses the same classification accuracy experiments show that these modified algorithms outperform their original counterparts it is worth noting that the performance of the modified algorithms becomes more visible when dealing with larger data sets

approaches to knowledge reduction of covering decision systems based on information theory in this paper we propose some new approaches for attribute reduction in covering decision systems from the viewpoint of information theory firstly we introduce information entropy and conditional entropy of the covering and define attribute reduction by means of conditional entropy in consistent covering decision systems secondly in inconsistent covering decision systems the limitary conditional entropy of the covering is proposed and attribute reductions are defined and finally by the significance of the covering some algorithms are designed to compute all the reducts of consistent and inconsistent covering decision systems we prove that their computational complexity are polynomial numerical tests show that the proposed attribute reductions accomplish better classification performance than those of traditional rough sets in addition in traditional rough set theory mibark algorithm g y wang h hu d yang decision table reduction based on conditional information entropy chinese j comput 25 2002 1 8 cannot ensure the reduct is the minimal attribute subset which keeps the decision rule invariant in inconsistent decision systems here we solve this problem in inconsistent covering decision systems

sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem a number of results exist in the literature for singularly perturbed differential equations without turning points in particular a number of difference schemes have been proposed that satisfy a stronger than normal convergence criteria known as uniform convergence this guarantees that the schemes model the boundary layers well we wish to examine whether these schemes will also be uniformly convergent if the equation has turning points to this end we derive sufficient conditions for uniform convergence which are satisfied not only by these schemes but by a more general class of schemes we show that the rate of convergence is determined by a characteristic parameter of the problem which may be less than one we confirm these theoretical results by numerical calculations

a priori estimates and analysis of a numerical method for a turning point problem bounds are obtained for the derivatives of the solution of a turning point problem these results suggest a modification of the el mistikawy werle finite difference scheme at the turning point a uniform error estimate is obtained for the resulting method and illustrative numerical results are given i introduction we will examine the following two point boundary value problem with dirichlet data at the endpoints l la ly eyx x p x yx x q x y x f x for i x 1

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