a donsker varadhan type of invariance principle letx 1 x 2 h be i i d random variables in the domain of attraction of a stable lawg and denote sn x1 xn ln a n 1 l n omega a n 1 sum limits j 0 n 1 chi a s j mathord left vphantom s j a n right kern nulldelimiterspace a n satisfiesa n 1 s n g large deviation probability estimates of donsker varadhan type are obtained forl n and these are then used to study the behavior of small values of s n a n these latter results are analogues of strassen s results which described the behavior of large values of s n a n when the limit law was gaussian the limiting constants are seen to depend only on the limit lawg and not on the distribution ofx 1 the techniques used are those developed by donsker and varadhan in their theory of large deviations

a matroidal approach to rough set theory rough set theory has been successfully applied to vague and uncertain data due to its approximation ability matroid is a sophisticated mathematical structure to provide a unifying abstract treatment for graph theory linear algebra and combinatorial optimization in this paper we redefine rough approximation operators through matroidal approaches and build a matroidal structure of rough set theory first each block of a partition is converted to a uniform matroid in this way a partition is transformed into a family of uniform matroids second these matroids are combined through the direct sum operation to form a new matroid therefore the scattered uniform matroids are treated as a whole one third each concept in a universe is transformed to a restriction matroid the lower and the upper approximations of each concept are established with the matroidal approach fourth for any two concepts in a universe the relationships between the approximations of them are discussed and some new properties are revealed these properties can be hardly found without the help of matroid theory fifth the boundary region and the negative region of a concept in the universe are established directly with the matroidal approach the lower and the upper approximations of each concept are then obtained through its boundary region this work indicates a new approach for studying rough set theory

quantitative analysis for covering based rough sets through the upper approximation number coverings are a useful form of data while covering based rough sets provide an effective tool for dealing with this data covering based rough sets have been widely used in attribute reduction and rule extraction however few quantitative analyses for covering based rough sets have been conducted while many advances for classical rough sets have been obtained through quantitative tools in this paper the upper approximation number is defined as a measurement to quantify covering based rough sets and a pair of upper and lower approximation operators are constructed using the approximation number the operators not only inherit some important properties of existing approximation operators but also exhibit some new quantitative characteristics it is interesting to note that the upper approximation number of a covering approximation space is similar to the dimension of a vector space or the rank of a matrix

on the structure of generalized rough sets in this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that 1 any reflexive binary relation determines a topology 2 if q is a reflexive and symmetric relation on a set x then o email protected x q a a is a topology such that a is open if and only if it is closed 3 conversely for every topological space x o satisfying the condition that a is open if and only if it is closed there exists a reflexive and symmetric relation r such that o email protected x r a a 4 let q be an equivalence relation on x for any pseudo w closed subset a of x q a is an w closed set if and only if w x x x email protected a for any email protected x moreover we consider properties of generalized rough sets

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

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

rules in incomplete information systems abstract a new method of computing all optimal certain rules from an incomplete information system is presented and proved the method does not require changing the size of the original incomplete system additionally several existing rough set methods of computing decision rules from incomplete information systems are analyzed and compared we show which of these methods are capable of generating all optimal certain rules or a class of optimal certain rules and which methods may lead to generation of false rules

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

large deviations for local time fractional brownian motion and applications abstract let w h w h t t r be a fractional brownian motion of hurst index h 0 1 with values in r and let l l t t 0 be the local time process at zero of a strictly stable levy process x x t t 0 of index 1 2 independent of w h the stable local time fractional brownian motion z h z h t t 0 is defined by z h t w h l t the process z h is self similar with self similarity index h 1 1 and is related to the scaling limit of a continuous time random walk with heavy tailed waiting times between jumps p becker kern m m meerschaert h p scheffler limit theorems for coupled continuous time random walks ann probab 32 2004 730 756 m m meerschaert h p scheffler limit theorems for continuous time random walks with infinite mean waiting times j appl probab 41 2004 623 638 however z h does not have stationary increments and is non gaussian in this paper we establish large deviation results for the process z h as applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for z h

an invariance principle for the local time of a recurrent random walk let s j be a lattice random walk i e s j x 1 x j where x 1 x 2 are independent random variables with values in the integer lattice and common distribution f and let l n omega k sum limits j 0 n 1 chi k s j omega the local time of the random walk at k before time n suppose ex 1 0 and f is in the domain of attraction of a stable law g of index 1 i e there exists a sequence a n necessarily of the form n 1 l n where l is slowly varying such that s n a n g define g n omega u frac c n n l n omega uc n where c n a n log log n and x greatest integer x then we identify the limit set of g n n 1 almost surely with a nonrandom set in terms of the i functional of donsker and varadhan the limit set is the one that donsker and varadhan obtain for the corresponding problem for a stable process several corollaries are then derived from this invariance principle which describe the asymptotic behavior of l n as n

a universal chung type law of the iterated logarithm let x 1 x 2 be a sequence of independent and identically distributed random variables we find sequences of norming and centering constants n and n such that a universal chung type law of the iterated logarithm holds namely lim inf n max 1 k n s k k n n almost surely where s k denotes the sum of the first k of x 1 x 2 k 1 if the underlying distribution function is in the feller class we show that this lim inf is strictly positive with probability 1

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

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

graphs with least eigenvalue 2 ten years on abstract the authors monograph spectral generalizations of line graphs was published in 2004 following the successful use of star complements to complete the classification of graphs with least eigenvalue 2 guided by citations of the book we survey progress in this area over the past decade some new observations are included

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

on almost sure limit inferior for b valued stochastic processes and applications a criterion on almost sure limit inferior for the increments of b valued stochastic processes is presented applications to processes of independent increments and to gaussian processes with stationary increments are given in particular an exact limit inferior bound is established for increments of infinite series of independent ornstein uhlenbeck processes

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

on the topological properties of fuzzy rough sets this paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe the tc axiom for fuzzy topology is proposed it is proved that the set of all lower approximation sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies tc axiom and conversely a fuzzy topology which obey tc axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation that is to say there exists a one to one correspondence between the set of all reflexive and transitive fuzzy relations and the set of all fuzzy topologies which satisfy tc axiom

the line graphs of lollipop graphs are determined by their spectra abstract haemers et al w h haemers x g liu y p zhang spectral characterizations of lollipop graphs linear algebra appl 428 2008 2415 2423 first investigated the spectral characterizations of the so called lollipop graphs h g l n with order n which is obtained by identifying a vertex of a cycle with order g and a pendent vertex of a path with order l for the graphs with least eigenvalue at least 2 cvetkovic and lepovic d cvetkovic m lepovic cospectral graphs with least eigenvalue at least 2 publ inst math nouv ser 78 92 2005 51 63 introduced the discriminant and the star value of a graph whose relations are investigated in this paper employing this relation and other techniques we prove that all line graphs of lollipop graphs are determined by their adjacency spectra

quick attribute reduct algorithm for neighborhood rough set model abstract in this paper we propose an efficient quick attribute reduct algorithm based on neighborhood rough set model in this algorithm we divide the objects records of the whole data set into a series of buckets based on their euclidean distances and then iterate each record by the sequence of buckets to calculate the positive region of neighborhood rough set model we also prove that each record s neighborhood elements can only be contained in its own bucket and its adjacent buckets thus it can reduce the iterations greatly based on the division of buckets we then present a new fast algorithm to calculate the positive region of neighborhood rough set model which can achieve a complexity of o m u m is the number of attributes u is the number of records containing in the data set furthermore with the new fast positive region computation algorithm we present a quick reduct algorithm for neighborhood rough set model and our algorithm can achieve a complexity of o m 2 u at last the efficiency of this quick reduct algorithm is proved by comparable experiments and especially this algorithm is more suitable for the reduction of big data

the small ball problem for the brownian sheet we show that the logarithm of the probability that the brownian sheet has a supremum at most e over 0 1 2 is of order e 2 log 1 e 3

on minimization of axiom sets characterizing covering based approximation operators rough set theory was proposed by pawlak to deal with the vagueness and granularity in information systems the classical relation based pawlak rough set theory has been extended to covering based generalized rough set theory the rough set axiom system is the foundation of the covering based generalized rough set theory because the axiomatic characterizations of covering based approximation operators guarantee the existence of coverings reproducing the operators in this paper the equivalent characterizations for the independent axiom sets of four types of covering based generalized rough sets are investigated and more refined axiom sets are presented

generalized fuzzy rough sets this paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized approximation operators is defined the connections between fuzzy relations and fuzzy rough approximation operators are examined in axiomatic approach various classes of fuzzy rough approximation operators are characterized by different sets of axioms axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators

nested structure in parameterized rough reduction abstract in this paper by strict mathematical reasoning we discover the relationship between the parameters and the reducts in parameterized rough reduction this relationship named the nested reduction shows that the reducts act as a nested structure with the monotonically increasing parameter we present a systematic theoretical framework that provides some basic principles for constructing the nested structure in parameterized rough reduction some specific parameterized rough set models in which the nested reduction can be constructed are pointed out by strict mathematical reasoning based on the nested reduction we design several quick algorithms to find a different reduct when one reduct is already given here different refers to the reducts obtained on the different parameters all these algorithms are helpful for quickly finding a proper reduct in the parameterized rough set models the numerical experiments demonstrate the feasibility and the effectiveness of the nested reduction approach

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

a note on the spectral characterization of θ graphs we consider graphs that is graphs obtained by subdividing the edges of the multigraph consisting of 3 parallel edges it is shown that any graph g is determined by the spectrum the multiset of eigenvalues except possibly when it contains a unique 4 cycle

probability estimates for multiparameter brownian processes soit f une fonction de repartition sur 0 1 d et w f le processus gaussien qui est la limite faible du processus empirique determine par f si g est une fonction sur 0 1 d on obtient des bornes superieures et inferieures pour p sup t 0 1 d w f t g t e

exceptional graphs with smallest eigenvalue 2 and related problems this paper summarizes the known results on graphs with smallest eigenvalue around 2 and completes the theory by proving a number of new results giving comprehensive tables of the finitely many exceptions and posing some new problems then the theory is applied to characterize a class of distance regular graphs of large diameter by their intersection array

large deviations for subordinated fractional brownian motion and applications abstract let w h w h t t r be a real valued fractional brownian motion with hurst index h 0 1 and let t t t t 0 be an inverse stable subordinator independent of w h the inverse stable subordinator fractional brownian motion z h z h t t 0 is defined by z h t w h t t which may arise as scaling limit of ctrw or random walk in a random environment in this paper we establish large deviation results for the process z h and its supremum process and we also give asymptotic properties of the tail probability of the supremum process

the lattice and matroid representations of definable sets in generalized rough sets based on relations abstract the definable set is a core concept in rough set theory it plays an important role in the characterizations of rough sets in this paper we study the lattice and matroid representations of definable sets in generalized rough sets based on relations first we propose the lower definable lattice consisting of all lower definable sets with set inclusion order then we give some conditions under which the lower definable lattice is distributive or geometric or boolean furthermore we discuss the relationship between distributive lattices and the lower definable lattices in generalized approximation spaces based on reflexive and transitive relations on the one hand we show that the lower definable lattice in a generalized approximation space based on reflexive and transitive relation is distributive on the other hand we obtain the result that a distributive lattice can induce a lower definable lattice in a generalized approximation space based on a reflexive and transitive relation finally we investigate the combination of generalized rough sets and matroids in terms of the lower definable lattice we show that if a lower definable lattice is a lattice of closed sets of a matroid then it must be an open closed set lattice of a matroid in addition we prove that some lower definable lattices are not the lattices of closed sets of matroids these results of this paper will benefit to our understanding of the relationship between matroids and generalized rough sets based on relations

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

on the signless laplacian spectral characterization of the line graphs of lollipop graphs abstract let h g k n denote the lollipop graph on n vertices obtained by identifying a vertex of the cycle c g of order g and an end vertex of the path p k 1 of order k 1 in this paper we prove that all line graphs l h g k n of lollipop graphs are determined by their laplacian spectrum furthermore we show that all l h g k n but the case that g k 3 are determined by their signless laplacian spectrum and also give the q cospectral class of l h r r n r 3

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

some properties of l fuzzy approximation spaces based on bounded integral residuated lattices in this paper we present a general framework for the study of l fuzzy rough sets based on bounded integral residuated lattices we introduce and discuss the notions of l fuzzy topological spaces left right lower and left right upper l fuzzy approximation operators based on a complete bounded integral residuated lattice investigate union intersection and composition operations of l fuzzy approximation spaces and study some properties of the l fuzzy topologies generated by these l fuzzy rough approximation operators and the l relations induced by an l fuzzy topology

continuity of l 2 valued ornstein uhlenbeck processes a stationary l 2 valued ornstein uhlenbeck process is considered which is given formally by dx t ax t dt 2a db t where a is a positive self adjoint operator on l 2 b t is a cylindrical brownian motion on l 2 and a is a positive diagonal operator on l 2 a simple criterion is given for the almost sure continuity of x t in l 2 which is shown to be quite sharp furthermore in certain special cases we obtain simple necessary and sufficient conditions for this almost sure continuity

neighborhood rough set based heterogeneous feature subset selection feature subset selection is viewed as an important preprocessing step for pattern recognition machine learning and data mining most of researches are focused on dealing with homogeneous feature selection namely numerical or categorical features in this paper we introduce a neighborhood rough set model to deal with the problem of heterogeneous feature subset selection as the classical rough set model can just be used to evaluate categorical features we generalize this model with neighborhood relations and introduce a neighborhood rough set model the proposed model will degrade to the classical one if we specify the size of neighborhood zero the neighborhood model is used to reduce numerical and categorical features by assigning different thresholds for different kinds of attributes in this model the sizes of the neighborhood lower and upper approximations of decisions reflect the discriminating capability of feature subsets the size of lower approximation is computed as the dependency between decision and condition attributes we use the neighborhood dependency to evaluate the significance of a subset of heterogeneous features and construct forward feature subset selection algorithms the proposed algorithms are compared with some classical techniques experimental results show that the neighborhood model based method is more flexible to deal with heterogeneous data

generalized rough sets based on relations rough set theory has been proposed by pawlak as a tool for dealing with the vagueness and granularity in information systems the core concepts of classical rough sets are lower and upper approximations based on equivalence relations this paper studies arbitrary binary relation based generalized rough sets in this setting a binary relation can generate a lower approximation operation and an upper approximation operation but some of common properties of classical lower and upper approximation operations are no longer satisfied we investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations

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

characteristic matrix of covering and its application to boolean matrix decomposition covering based rough sets provide an efficient means of dealing with covering data which occur widely in practical applications boolean matrix decomposition has frequently been applied to data mining and machine learning in this paper three types of existing covering approximation operators are represented by boolean matrices and then used in boolean matrix decomposition first we define two characteristic matrices of a covering through these boolean characteristic matrices three types of existing covering approximation operator are concisely and equivalently represented second these operator representations are applied to boolean matrix decomposition which has a close relationship with nonnegative matrix factorization a popular and efficient technique for machine learning we provide a sufficient and necessary condition for a square boolean matrix to decompose into the boolean product of another matrix and its transpose we then develop an algorithm for this boolean matrix decomposition finally these three covering approximation operators are axiomatized using boolean matrices this work presents an interesting viewpoint from which to investigate covering based rough set theory and its applications

catoptrical rough set model on two universes using granule based definition and its variable precision extensions rough set models on two universes are valuable generalizations of classic rough set model these models can be applied into formal concept analysis and decision making system this paper focuses on catoptrical approach for rough set models over two universes using granule based definition in this paper we propose a new catoptrical rough set model over dual universes the properties of the proposed model are discussed consequently a variable precision extension of the model is presented some properties of the extended model are obtained finally we investigate a further extension of the proposed model i e variable precision catoptrical rough set model with two parameters

analysis of a stochastic implicit interface model for an immersed elastic surface in a fluctuating fluid we present some mathematical analyses of a recently proposed stochastic implicit interface model for an elastic surface immersed in an incompressible viscous fluid subject to fluctuation forces we derive suitable a priori estimates and establish the well posedness of pathwise solutions and provide uniform control on the solutions in probability

path properties of kernel generated two time parameter gaussian processes we study path properties of two parameter gaussian processes x t v t r of the formx t v 0 t v x y dw x y where the kernel function t v x y is assumed to be square integrable in x y onr r andw x y is a standard two parameter wiener process

the rough membership functions on four types of covering based rough sets and their applications abstract pawlak s rough membership functions not only give numerical characterizations of pawlak s rough set approximations but also establishes the relationships between pawlak s rough sets and fuzzy sets or probabilistic rough sets however it is noteworthy that pawlak s rough membership functions have limitations when handling incomplete data that exist widely in the real world as will be shown in this paper one way to overcome this is to construct rough membership functions for covering based rough sets in this paper we first use an example in evidence based medicine to illustrate how to use pawlak s rough membership function on numerically characterizing decisions under the circumstances where data are complete then we construct covering based rough membership functions for four types of covering based rough sets which were examined by zhu and wang in ieee transactions on knowledge and data engineering 19 8 2007 1131 1144 and information sciences 201 2012 80 92 and use them to characterize these covering based rough sets numerically finally we present theoretical backgrounds for these covering based rough membership functions and illustrate how to apply them on numerically characterizing decisions under the circumstances where data are incomplete

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

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

metric entropy and the small ball problem for gaussian measures abstract we establish a precise link between the small ball problem for a gaussian measure on a separable banach space and the metric entropy of the unit ball of the hubert space h generating this link allows us to compute small ball probabilities from metric entropy results and vice versa

whittaker kotel nikov shannon approximation of φ sub gaussian random processes abstract the article starts with generalizations of some classical results and new truncation error upper bounds in the sampling theorem for bandlimited stochastic processes then it investigates l p 0 t and uniform approximations of sub gaussian random processes by finite time sampling sums explicit truncation error upper bounds are established some specifications of the general results for which the assumptions can be easily verified are given direct analytical methods are employed to obtain the results

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

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

comments to the line graphs of lollipop graphs are determined by their spectra abstract in j f wang and s n shu 2012 2 the authors lost the condition that the line graph l g l n does not contain the cycle of order 4 in lemma 3 4 which makes their main result i e theorem 3 1 of that paper not completely correct here some comments are given

functional limit theorems for the infinite series of ou processes in holder norm in this paper estimating large deviation probabilities for the infinite series of independent ou processes in holder norm we obtain functional limit theorems for this process in the holder norm which include functional modulus of continuity and functional lil for the infinite series of independent ou processes in the holder norm

minima of h valued gaussian processes we study low local extremes of gaussian random fields with values in a separable hilbert space and constant variance our results are sharp for certain stationary processes on the line and for these processes we also prove global limits

extended results on the relationship between information systems the relationship between information systems is an important topic in rough set field and it is studied through mappings many kinds of consistent functions have been proposed and they can be used to describe the invariance of attribute reducts between two information systems efficiently however they cannot reflect relationships between neighborhoods generated from two information systems in this paper to achieve this aim firstly we introduce a neighborhood continuous function that is inspired by the concept of continuous function in topology then we address its properties and discuss relationships between neighborhood continuous functions and several kinds of existing consistent functions furthermore we investigate some important properties of neighborhood continuous function with respect to relation mappings finally based on neighborhood continuous functions the notion of a neighborhood homomorphism between different information systems is proposed it is noted that the reduct feature of a single information system can be described by neighborhood continuous functions while the reduct invariance of two different information systems can be described by neighborhood homomorphisms

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

small ball probabilities of gaussian fields lower bounds on the small ball probability are given for brownian sheet type gaussian fields as well as for general gaussian fields with stationary increments in d in particular a sharp bound is found for the fractional levy brownian fields

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

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

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

spectral characterization of graphs with index at most 2 5 abstract a graph is said to be determined by the adjacency spectrum ds for short if there is no other nonisomorphic graph with the same spectrum all connected graphs with index at most 2 5 are known in this paper we show that with few exceptions all of these graphs are ds

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

generalized rough sets over fuzzy lattices this paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches from the viewpoint of the constructive approach the basic properties of generalized rough sets over fuzzy lattices are obtained the matrix representation of the lower and upper approximations is given according to this matrix view a simple algorithm is obtained for computing the lower and upper approximations as for the axiomatic approach a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices

on the union and intersection operations of rough sets based on various approximation spaces algebraic structures and lattice structures of rough sets are basic and important topics in rough sets theory in this paper we pointed out that a basic problem had been overlooked that is the closeness of union and intersection operations of rough approximation pairs i e lower approximation upper approximation we present that the union and intersection operations of rough approximation pairs are closed for classical rough sets and two kinds of covering based rough sets but not for twenty kinds of covering based rough sets and the generalized rough sets based on fuzzy approximation space moreover we proved that the union and intersection operations of rough fuzzy approximation pairs are closed and a bounded distributive lattice can be constructed

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

spectral characterizations of lollipop graphs abstract the lollipop graph denoted by h n p is obtained by appending a cycle c p to a pendant vertex of a path p n p we will show that no two non isomorphic lollipop graphs are cospectral with respect to the adjacency matrix it is proved that for p odd the lollipop graphs h n p and some related graphs h n p are determined by the adjacency spectrum and that all lollipop graphs are determined by its laplacian spectrum

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

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

on the continuity of ornstein uhlenbeck processes in infinite dimensions existence and continuity of ornstein uhlenbeck processes in banach and hilbert spaces are investigated under various assumptions

on the small time asymptotics of diffusion processes on hilbert spaces in this paper we establish a small time large deviation principle and obtain the following small time asymptotics lim t 0 2t log p x 0 b x t c d 2 b c for diffusion processes on hilbert spaces where d b c is the intrinsic metric between two subsets b and c associated with the diffusions the case of perturbed ornstein uhlenbeck processes is treated separately at the end of the paper

knowledge acquisition in incomplete information systems a rough set approach abstract this paper deals with knowledge acquisition in incomplete information systems using rough set theory the concept of similarity classes in incomplete information systems is first proposed two kinds of partitions lower and upper approximations are then formed for the mining of certain and association rules in incomplete decision tables one type of optimal certain and two types of optimal association decision rules are generated two new quantitative measures random certainty factor and random coverage factor associated with each decision rule are further proposed to explain relationships between the condition and decision parts of a rule in incomplete decision tables the reduction of descriptors and induction of optimal rules in such tables are also examined