rough set theory applied to fuzzy ideal theory we use covers of the universal set to define approximation operators on the power set of the given set in section 1 we determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural properties of certain subsets we define a particular cover on the set of ideals of a commutative ring with identity in such a way that both the concepts of the fuzzy prime spectrum of a ring and rough set theory can simultaneously be brought to bear on the study of fuzzy ideals of a ring

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

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

the ϑ lower and t upper fuzzy rough approximation operators on a semigroup in this paper we consider a semigroup as a universal set and investigate the properties of q lower and t upper fuzzy rough approximation operators a new fuzzy algebraic structure called a tl fuzzy rough semigroup is introduced and some related properties are studied finally the homomorphism properties of q lower and t upper fuzzy rough approximation operators are investigated

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

filtered backprojection and the exponential radon transform on etudie des conditions sur la mesure qui assurent la validite de 7b r t 7b rf k f e ou e 7b r t k pour une transformee de radon generalisee 7b r et une grande classe de filtres k correspondant a des fonctions delta approchees e

some results on fuzzy prime spectrum of a ring abstract in this note we extend the results of kumar 1992 in this context we first give a counter example to his theorem 3 4 ii and then we prove that the fuzzy prime spectrum of a ring and the elements of its basis are both compact finally by giving two examples we will show that it is not true in general that any element of a basis of the fuzzy prime spectrum of a boolean ring is closed and the fuzzy prime spectrum itself is hausdorff also by an example it is shown that the proof of the necessity of the condition of theorem 5 3 of kumar 1992 is incorrect

computerized tomography with unknown sources we give a reconstruction method for the attenuation coefficient in a 2d cross section of a 3d object from tomographic data which does not require knowledge of the positions or the intensity of the sources and the sources may even be inside the object such a situation arises in emission computerized tomography the method is based on consistency conditions in the range of the relevant integral transforms the paper contains a detailed description of the algorithm and numerical results for computer generated data

a random map implementation of implicit filters implicit particle filters for data assimilation generate high probability samples by representing each particle location as a separate function of a common reference variable this representation requires that a certain underdetermined equation be solved for each particle and at each time an observation becomes available we present a new implementation of implicit filters in which we find the solution of the equation via a random map as examples we assimilate data for a stochastically driven lorenz system with sparse observations and for a stochastic kuramoto sivashinsky equation with observations that are sparse in both space and time

reconstruction from uniformly attenuated spect projection data using the dbh method an algorithm was developed for the 2 d reconstruction of truncated and nontruncated uniformly attenuated data acquired from single photon emission computed tomography spect the algorithm is able to reconstruct data from half scan 180deg and short scan 180deg fan angle acquisitions for parallel and fan beam geometries respectively as well as data from full scan 360deg acquisitions the algorithm is a derivative backprojection and hilbert transform dbh method which involves the backprojection of differentiated projection data followed by an inversion of the finite weighted hilbert transform the kernel of the inverse weighted hilbert transform is solved numerically using matrix inversion numerical simulations confirm that the dbh method provides accurate reconstructions from half scan and short scan data even when there is truncation however as the attenuation increases finer data sampling is required

some properties of generalized rough sets some algebraic and topological properties of generalized rough sets associated with generalized lower and upper approximations are studied generalized definable sets give rise to two topological spaces degree of accuracy dag for generalized rough sets induces two types of equivalence relations one is defined on x whereas the other is defined on p y both of these relations strictly manage an order in classes of x and p y it has been shown that dag induces two types of fuzzy subsets one of the set x and the other of p y finally some properties of these fuzzy sets have been discussed

some comments on spectrums of prime fuzzy ideals of a ring abstract common and distinctive features of two prime spectrums one obtained by kumar and kohli using 2 valued prime fuzzy ideals and the other obtained by the author using not so restrictive definition of prime fuzzy ideals are discussed it is also shown that some results claimed by kumar and kohli regarding spectrum of 2 valued prime fuzzy ideals need some corrections

rough sets induced by ideals in lattices this paper is to study the rough sets within the context of lattices we study the special properties of the rough sets which can be constructed by means of the congruences determined by ideals of lattice also the properties of the generalized rough sets with respect to ideals of lattice are investigated finally we give an example of their application in formal concept analysis

on fuzzy ideals of a ring 1 abstract the aim of this paper is to study fuzzy ideals of a ring to characterise regular rings and noetherian rings by fuzzy ideals and to determine all fuzzy prime ideals of the ring z of integers

semi monolayer cover rough set concept property and granular algorithm abstract rough set theory is known as a powerful tool for dealing with incomplete inconsistent and insufficient information in decision making with the constraint that each set of cover has at least one reliable element the semi monolayer cover smc is a special neighborhood system defined on the universe between the partition and regular cover more similar to pawlak approximation operators than cover approximation operators smc approximation operators have some interesting properties such as the equivalent granular forms the cross duality etc in this paper we define four pairs of smc rough approximation operators namely gc gu de and da and then discuss their properties and relations it has been proven that 1 gu is the equivalent granular form approximation operator of de 2 de and da are two pairs of cross dual approximation operators that is to say that upper de and lower da lower de and upper da are two pairs of dual approximation operators the equivalent relation and cross dual relation serve as a bridge connecting granular form approximation operators with the dot form ones 3 a de or da approximation set can be rebuilt by the corresponding gc approximation set and an smc mapping the equivalent granular form of de and da gives rise to some advanced algorithms that improve the efficiency of calculating de and da approximation sets experimental results on a uci data set show the granular algorithm achieves better performance than the basic dot algorithm and the speedup of efficiency tends to be exponential with the size of universe

roughness based on fuzzy ideals the theory of rough set proposed by pawlak and the theory of fuzzy set proposed by zadeh are complementary generalizations of classical set theory many sets are naturally endowed with two binary operations addition and multiplication one concept which does this is a ring this paper concerns a relationship between rough sets fuzzy sets and ring theory it is a continuation of ideas presented by kuroki and wang n kuroki p p wang the lower and upper approximations in a fuzzy group inform sci 90 1996 203 220 we consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal in fact we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring some characterizations of the above approximations are made and some examples are presented

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

communication between information systems with covering based rough sets communication between information systems is considered as an important issue in granular computing the concept of homomorphism is an effective mathematical tool to study information exchange between information systems this paper provides a study on some basic properties of covering information systems and decision systems under homomorphisms first we define consistent functions related to coverings and covering mappings between two universes and study their properties then we introduce the notions of homomorphisms of covering information systems and point out that a homomorphism is a special covering mapping between information systems furthermore we investigate some important properties of homomorphisms in covering information systems and decision systems it is proved that some basic properties of original systems such as set approximations attribute reductions can be reserved under the condition of homomorphisms in both covering information systems and covering decision systems

roughness and fuzziness in quantales abstract rough set theory and fuzzy set theory are two useful mathematical tools for dealing with uncertainty and granularity in information systems motivated by the studies of roughness and fuzziness in algebraic systems and partially ordered sets such as semigroups rings and lattices in this paper we introduce first the notions of fuzzy prime semi prime primary ideals of quantales and investigate their properties several characterizations of such ideals are presented then we introduce the concepts of weak fuzzy prime ideals and strong fuzzy prime ideals of quantales and establish relationships between these ideals and fuzzy prime ideals of quantales by applying rough set theory to fuzzy ideals of quantales we furthermore define rough fuzzy prime semi prime primary ideals of quantales generalizing yang and xu s work on quantales to the fuzzy environment finally relationships between the upper resp lower rough fuzzy ideals of quantales and the upper resp lower approximations of their homomorphic images are also discussed

regularization approach for inverting the exponential radon transforms ert in this paper we study the problem of inverting the exponential radon transforms with non exact data a regularization approach is introduced to find approximate solutions of the problem

roughness in modules the theory of rough sets was proposed by pawlak in 1982 the relations between rough sets and algebraic systems have been already considered by many mathematicians important algebraic structures are groups rings and modules rough groups and rough rings have been investigated by biswas and nanda kuroki and wang and davvaz in this paper we consider an r module as a universal set and we introduce the notion of rough submodule with respect to a submodule of an r module which is an extended notion of a submodule in an r module we also give some properties of the lower and the upper approximations in an r module

spectrum of prime fuzzy ideals abstract a topological space called a spectrum of prime fuzzy ideals and denoted by fspec r is obtained by introducing a topology on the set of prime fuzzy ideals of a ring r the prime fuzzy ideals used here introduced by the author elsewhere are not necessarily two valued the topology introduced is compact and the usual nonfuzzy prime spectrum spec r is a subspace of fspec r and dense in it situations under which fuzzy spectrums of different rings are homeomorphic are discussed in particular fspec r is proved to be homeomorphic to fspec r frad r where frad r is the fuzzy prime radical of r the correspondence associating a ring r to the topological space fspec r is shown to define a contravarient functor from the category of rings with unity into the category of compact topological spaces

on rough set and fuzzy sublattice let l be a lattice with the least element 0 and the greatest element 1 and let be a full congruence relation on l in this paper the notion of upper and lower approximations of a fuzzy subset of l is introduced and some important properties will be studied

the lower and upper approximations in a quotient hypermodule with respect to fuzzy sets this paper provides a continuation of ideas presented by davvaz and mahdavipour b davvaz m mahdavipour roughness in modules inform sci 176 2006 3658 3674 the notion of hypermodule is a generalization of the notion of module in this paper we consider the quotient hypermodule m a and interpret the lower and upper approximations as subsets of the quotient hypermodule m a then we introduce the concept of quotient rough sub hypermodule also using the concept of fuzzy sets we introduce and discuss the concept of fuzzy rough hypermodules and then we obtain the relation between fuzzy rough sub hypermodules and level rough sets this relation is characterized as a necessary and sufficient condition

accelerated monte carlo for optimal estimation of time series by casting stochastic optimal estimation of time series in path integral form one can apply analytical and computational techniques of equilibrium statistical mechanics in particular one can use standard or accelerated monte carlo methods for smoothing filtering and or prediction here we demonstrate the applicability and efficiency of generalized nonlocal hybrid monte carlo and multigrid methods applied to optimal estimation specifically smoothing we test these methods on a stochastic diffusion dynamics in a bistable potential this particular problem has been chosen to illustrate the speedup due to the nonlocal sampling technique and because there is an available optimal solution which can be used to validate the solution via the hybrid monte carlo strategy in addition to showing that the nonlocal hybrid monte carlo is statistically accurate we demonstrate a significant speedup compared with other strategies thus making it a practical alternative to smoothing filtering and data assimilation on problems with state vectors of fairly large dimensions as well as a large total number of time steps

representation theoretic patterns in three dimensional cryo electron microscopy i the intrinsic reconstitution algorithm in this paper we reveal the formal algebraic structure underlying the intrinsic reconstitution algorithm introduced by singer and shkolnisky in 9 for determining three dimensional macromolecular structures from images obtained by an electron microscope inspecting this algebraic structure we obtain a conceptual explanation for the admissibility correctness of the algorithm and a proof of its numerical stability in addition we explain how the various numerical observations reported in that work follow from basic representation theoretic principles

the fourth type of covering based rough sets as a technique for granular computing rough sets deal with the vagueness and granularity in information systems covering based rough sets have been proposed to generalize this theory for wider application three types of covering based rough sets have been studied for different situations to make the theory more complete this paper proposes a fourth type of covering based rough sets compared with the existing ones the new type shows its special characteristic in the interdependency between its lower and upper approximations we carry out a systematical study of this new theory first we discuss basic properties such as normality contraction and monotone then we investigate the conditions for this type of covering based rough sets to satisfy the properties of pawlak s rough sets and study the interdependency between the lower and upper approximation operations in addition axiomatic systems for the lower and upper approximation operations are established lastly we address the relationships between this type of covering based rough sets and the three existing ones

θ t fuzzy rough approximation operators and the tl fuzzy rough ideals on a ring in this paper we consider a ring as a universal set and study t fuzzy rough approximation operators with respect to a tl fuzzy ideal of a ring first some new properties of generalized t fuzzy rough approximation operators are obtained then a new fuzzy algebraic structure tl fuzzy rough ideal is defined and its properties investigated and finally the homomorphism of t fuzzy rough approximation operators is studied

measures of general fuzzy rough sets on a probabilistic space fuzzy rough set is a generalization of crisp rough set which deals with both fuzziness and vagueness in data the measures of fuzzy rough sets aim to dig its numeral characters in order to analyze data effectively in this paper we first develop a method to compute the cardinality of fuzzy set on a probabilistic space and then propose a real number valued function for each approximation operator of the general fuzzy rough sets on a probabilistic space to measure its approximate accuracy the functions of lower and upper approximation operators are natural generalizations of the belief function and plausibility function in dempster shafer theory of evidence respectively by using these functions accuracy measure roughness degree dependency function entropy and conditional entropy of general fuzzy rough set are proposed and the relative reduction of fuzzy decision system is also developed by using the dependency function and characterized by the conditional entropy at last these measure functions for approximation operators are characterized by axiomatic approaches

fuzzy prime spectrum of a ring ii abstract various topological properties of the fuzzy prime spectrum of a ring and certain of its subspaces as well as its functorial nature are discussed

fuzzy prime ideals of a ring abstract this paper characterizes all fuzzy prime ideals p of an arbitrary ring r we show that a nonconstant fuzzy ideal p of r is prime if and only if p0 x r p x p 0 is a prime ideal of r p is two valued and p 0 1 examples are given showing that p0 is a prime ideal is not sufficient for p to be a fuzzy prime ideal and that p0 may be a prime ideal of r with the image of p being of arbitrary finite cardinality or being countably infinite

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

fuzzy rough approximations for set valued data rough set theory is one of important tools of soft computing and rough approximations are the essential elements in rough set models however the existing fuzzy rough set model for set valued data which is directly constructed based on a kind of similarity relation fail to explicitly define fuzzy rough approximations to solve this issue in this paper we propose two types of fuzzy rough approximations and define two corresponding relative positive region reducts furthermore two discernibility matrices and two discernibility functions are introduced to acquire these new proposed reducts and the relationships among the new reducts and the existing reducts are also be provided theoretical analyses demonstrate that the new types of reducts have less redundancy and are more diverse no lower number of reducts than those obtained by means of the existing matrices and experimental results illustrate the new reducts found by our methods outperform those obtained by existing method

rough set theory applied to lattice theory in this paper we intend to study a connection between rough sets and lattice theory we introduce the concepts of upper and lower rough ideals filters in a lattice then we offer some of their properties with regard to prime ideals filters the set of all fixed points compact elements and homomorphisms

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

rough sets based on complete completely distributive lattice in this paper a pair of rough approximation operators on a complete completely distributive ccd lattice based on an ordinary binary relation is defined this kind of rough sets can be seen as a unified framework for the study of rough sets based on ordinary binary relations rough fuzzy sets and interval valued rough fuzzy set moreover depending on classes of binary relations this paper defines several classes of rough sets on ccd lattices and investigates properties of these classes finally two generalized rough set models on two ccd lattices are given at the end of this paper

minimization of axiom sets on fuzzy approximation operators axiomatic characterization of approximation operators is an important aspect in the study of rough set theory in this paper we examine the independence of axioms and present the minimal axiom sets characterizing fuzzy rough approximation operators and rough fuzzy approximation operators

implicit sampling for particle filters we present a particle based nonlinear filtering scheme related to recent work on chainless monte carlo designed to focus particle paths sharply so that fewer particles are required the main features of the scheme are a representation of each new probability density function by means of a set of functions of gaussian variables a distinct function for each particle and step and a resampling based on normalization factors and jacobians the construction is demonstrated on a standard ill conditioned test problem

particle filtering with path sampling and an application to a bimodal ocean current model this paper introduces a recursive particle filtering algorithm designed to filter high dimensional systems with complicated non linear and non gaussian effects the method incorporates a parallel marginalization pmmc step in conjunction with the hybrid monte carlo hmc scheme to improve samples generated by standard particle filters parallel marginalization is an efficient markov chain monte carlo mcmc strategy that uses lower dimensional approximate marginal distributions of the target distribution to accelerate equilibration as a validation the algorithm is tested on a 2516 dimensional bimodal stochastic model motivated by the kuroshio current that runs along the japanese coast the results of this test indicate that the method is an attractive alternative for problems that require the generality of a particle filter but have been inaccessible due to the limitations of standard particle filtering strategies

following a moving target monte carlo inference for dynamic bayesian models markov chain monte carlo mcmc sampling is a numerically intensive simulation technique which has greatly improved the practicality of bayesian inference and prediction however mcmc sampling is too slow to be of practical use in problems involving a large number of posterior target distributions as in dynamic modelling and predictive model selection alternative simulation techniques for tracking moving target distributions known as particle filters which combine importance sampling importance resampling and mcmc sampling tend to suffer from a progressive degeneration as the target sequence evolves we propose a new technique based on these same simulation methodologies which does not suffer from this progressive degeneration

improved particle filters for multi target tracking we present a novel approach for improving particle filters for multi target tracking the suggested approach is based on drift homotopy for stochastic differential equations drift homotopy is used to design a markov chain monte carlo step which is appended to the particle filter and aims to bring the particle filter samples closer to the observations while at the same time respecting the target dynamics we have used the proposed approach on the problem of multi target tracking with a nonlinear observation model the numerical results show that the suggested approach can improve significantly the performance of a particle filter

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 systematic study on attribute reduction with rough sets based on general binary relations attribute reduction is considered as an important preprocessing step for pattern recognition machine learning and data mining this paper provides a systematic study on attribute reduction with rough sets based on general binary relations we define a relation information system a consistent relation decision system and a relation decision system and their attribute reductions furthermore we present a judgment theorem and a discernibility matrix associated with attribute reduction in each type of system based on the discernibility matrix we can compute all the reducts finally the experimental results with uci data sets show that the proposed reduction methods are an effective technique to deal with complex data sets

semantic analysis of inductive reasoning abstract inductive learning is analysed from the semantic point of view processes of forming generalisations of concepts determined by examples of expert decisions are discussed it is claimed that since concepts are relative to background knowledge their inductive generalisations can be determined only approximately induction rules are defined providing a method of forming generalisations preserving positive and or negative instances of concepts

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

two new operators in rough set theory with applications to fuzzy sets in this paper two new operators are introduced for the rough set theory using them two inequalities well known in the rough set theory can now be modified to become equalities with this change no information will be lost in the new expressions hence many properties in rough set theory can be improved and in particular the union the intersection and the complement operations can be redefined based on the two equalities furthermore the collection of rough sets of an approximation space forms a boolean algebra under these new operators finally roughness properties of fuzzy sets are analyzed using the new operations

generalized lower and upper approximations in a ring in this paper the concepts of set valued homomorphism and strong set valued homomorphism of a ring are introduced and related properties are investigated the notions of generalized lower and upper approximation operators constructed by means of a set valued mapping which is a generalization of the notion of lower and upper approximation of a ring are provided we also propose the notion of generalized lower and upper approximations with respect to an ideal of a ring which is an extended notation of rough ideal introduced lately by davvaz b davvaz roughness in rings information science 164 2004 147 163 in a ring and discuss some significant properties of them

axiomatic approaches to rough approximation operators on complete completely distributive lattices we proposed a pair of rough approximation operators on a complete completely distributive lattice ccd lattice for short in 2015 in this paper we further discuss its properties and study the axiomatic approaches to the rough approximation operators through these axioms fuzzy rough approximation operators can be seen as special cases of rough approximation operators on a ccd lattice we also discuss the axiomatic approaches to generalized rough sets on ccd lattices

the minimization of axiom sets characterizing generalized approximation operators in the axiomatic approach of rough set theory rough approximation operators are characterized by a set of axioms that guarantees the existence of certain types of binary relations reproducing the operators thus axiomatic characterization of rough approximation operators is an important aspect in the study of rough set theory in this paper the independence of axioms of generalized crisp approximation operators is investigated and their minimal sets of axioms are presented

variance reduction for particle filters of systems with time scale separation we present a particle filter construction for a system that exhibits time scale separation the separation of time scales allows two simplifications that we exploit 1 the use of the averaging principle for the dimensional reduction of the dynamics for each particle during the prediction step and 2 the factorization of the transition probability for the rao blackwellization of the update step the resulting particle filter is faster and has smaller variance than the particle filter based on the original system the method is tested on a multiscale stochastic differential equation and on a multiscale pure jump diffusion motivated by chemical reactions

generalized fuzzy rough sets determined by a triangular norm the theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications various fuzzy generalizations of rough approximations have been made over the years this paper presents a general framework for the study of t fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used by using a pair of dual triangular norms in the constructive approach some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations the connections between special fuzzy relations and the t upper and t lower approximation operators of fuzzy sets are also examined in the axiomatic approach an operator oriented characterization of rough sets is proposed that is t fuzzy approximation operators are defined by axioms different axiom sets of t upper and t lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators the independence of axioms characterizing the t fuzzy rough approximation operators is examined then the minimal sets of axioms for the characterization of the t fuzzy approximation operators are presented based on information theory the entropy of the generalized fuzzy approximation space which is similar to shannon s entropy is formulated to measure uncertainty in t generalized fuzzy rough sets a notion of fuzziness is introduced some basic properties of this measure are examined for a special triangular norm t min it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable

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