rough support vector regression abstract this paper describes the relationship between support vector regression svr and rough or interval patterns svr is the prediction component of the support vector techniques rough patterns are based on the notion of rough values which consist of upper and lower bounds and are used to effectively represent a range of variable values predictions of rough values in a variety of different forms within the context of interval algebra and fuzzy theory are attracting research interest an extension of svr called rough support vector regression rsvr is proposed to improve the modeling of rough patterns in particular it is argued that the upper and lower bounds should be modeled separately the proposal is shown to be a more flexible version of lower possibilistic regression model using insensitivity experimental results on the dow jones industrial average demonstrate the suggested rsvr modeling technique

textural unit operations in rough set theory abstract in this study we introduce a textural counterpart of the unit operation of wybraniec skardowska a unit di operation has two parts which are called a unit co operation and a unit operation respectively in this respect we have two types of symmetry we show that the symmetricity of direlations is equivalent to the galois connectivity of unit di operations for discrete textures we determine the image co operation of a unit co operation we prove that the symmetricity and duality are equivalent concepts for unit di operations if one of the compounds is symmetric we consider definability in terms of unit operations and unit co operations further we present a categorical discussion defining a category un whose structures of objects are unit operations we show that the category rel whose objects are approximation spaces and morphisms are relation preserving functions can be embedded into un

spectral asymptotics for laplacians on self similar sets abstract given a self similar dirichlet form on a self similar set we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a geometric counting function defined through a family of coverings of the self similar set naturally associated with the dirichlet space secondly under sub gaussian heat kernel upper bound we prove a detailed short time asymptotic behavior of the partition function which is the laplace stieltjes transform of the eigenvalue counting function associated with the dirichlet form this result can be applicable to a class of infinitely ramified self similar sets including generalized sierpinski carpets and is an extension of the result given recently by b m hambly for the brownian motion on generalized sierpinski carpets moreover we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

the relationship among different covering approximations this paper studies covering based rough sets by means of characteristic functions of sets we consider covering based rough sets from both constructive and axiomatic approaches the relationship among four types of covering based rough sets is discussed we also outline the topologies induced by different covering approximations

hardy littlewood sobolev inequalities via fast diffusion flows we give a simple proof of the d 2 cases of the sharp hardy littlewood sobolev inequality for d 3 and the sharp logarithmic hardy littlewood sobolev inequality for d 2 via a monotone flow governed by the fast diffusion equation

a comparative study of fuzzy sets and rough sets this paper reviews and compares theories of fuzzy sets and rough sets two approaches for the formulation of fuzzy sets are reviewed one is based on many valued logic and the other is based on modal logic two views of rough sets are presented set oriented view and operator oriented view rough sets under set oriented view are closely related to fuzzy sets which leads to non truth functional fuzzy set operators both of them may be considered as deviations of classical set algebra in contrast rough sets under operator oriented view are different from fuzzy sets and may be regarded as an extension of classical set algebra

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

a mating of trees approach for graph distances in random planar maps we introduce a general technique for proving estimates for certain random planar maps which belong to the gamma liouville quantum gravity lqg universality class for gamma in 0 2 the family of random planar maps we consider are those which can be encoded by a two dimensional random walk with i i d increments via a mating of trees bijection and includes the uniform infinite planar triangulation uipt gamma sqrt 8 3 and planar maps weighted by the number of different spanning trees gamma sqrt 2 bipolar orientations gamma sqrt 4 3 or schnyder woods gamma 1 that can be put on the map using our technique we prove estimates for graph distances in the above family of random planar maps in particular we obtain non trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the watabiki prog theor phys suppl 114 1 17 1993 prediction for the hausdorff dimension of gamma lqg and we establish the existence of an exponent for certain distances in the map the basic idea of our approach is to compare a given random planar map m to a mated crt map a random planar map constructed from a correlated two dimensional brownian motion using a strong coupling zaitsev in esaim probab stat 2 41 108 1998 of the encoding walk for m and the brownian motion used to construct the mated crt map this allows us to deduce estimates for graph distances in m from the estimates for graph distances in the mated crt map which we proved using continuum theory in a previous work in the special case when gamma sqrt 8 3 we instead deduce estimates for the sqrt 8 3 mated crt map from known results for the uipt the arguments of this paper do not directly use sle lqg and can be read without any knowledge of these objects

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

a sequential formulation of a logic based on fuzzy modalities abstract a nakamura proposed a logic based on fuzzy modalities the main purpose of this paper is to present a gentzen type sequential formulation of this logic

rough set theory for the interval valued fuzzy information systems the notion of a rough set was originally proposed by pawlak z pawlak rough sets international journal of computer and information sciences 11 5 1982 341 356 later on dubois and prade d dubois h prade rough fuzzy sets and fuzzy rough sets international journal of general system 17 2 3 1990 191 209 introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets this paper deals with an interval valued fuzzy information system by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory and discusses the basic rough set theory for the interval valued fuzzy information systems in this paper we firstly define the rough approximation of an interval valued fuzzy set on the universe u in the classical pawlak approximation space and the generalized approximation space respectively i e the space on which the interval valued rough fuzzy set model is built secondly several interesting properties of the approximation operators are examined and the interrelationships of the interval valued rough fuzzy set models in the classical pawlak approximation space and the generalized approximation space are investigated thirdly we discuss the attribute reduction of the interval valued fuzzy information systems finally the methods of the knowledge discovery for the interval valued fuzzy information systems are presented with an example

on relationship between modified sets topological spaces and rough sets abstract in this paper we define modifiers by relations especially weakening and substantiating modifiers are defined by a so called accessibility relation which is a reflexive relation on a non empty set x after presenting some main results we prove that this type of modifiers will satisfy the kuratowski closure axioms this means that modifiers in fact induce topological spaces also rough sets are considered to be a special case of modified sets

on the topological properties of generalized rough sets in this paper we consider some topological properties of generalized rough sets induced by binary relations and show that1 any serial binary relation can induce a topology 2 let r be a binary relation on a universe u t r and e r denote the transitive closure and the equivalence closure of r respectively if r is a reflexive relation on u then r and t r induce the same topology i e t r t t r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators t r and t r respectively moreover r t r t r where r t r is the relation induced by the topology t r 3 when r is a reflexive and symmetric relation r and e r induce the same topology i e t r t e r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators e r and e r respectively moreover r t r e r 4 based on the above conclusions the notion of topological reduction of incomplete information systems is proposed and characterizations of reduction of consistent incomplete decision tables are obtained

asymptotic behaviour near finite time extinction for the fast diffusion equation we study the cauchy problem for the fast diffusion equation u t delta u m u gee 0 qquad hbox in bf r n times bf r when n gee 3 and 0 m n 2 over n for a class of nonnegative radially symmetric finite mass solutions which vanish identically at a given time t 0 we show that their asymptotic behaviour as t nearrow t is described by a uniquely determined self similar solution of the second kind u r t t t alpha f eta where eta r t t beta and alpha 1 2 beta over 1 m and r x here beta is determined from a nonlinear eigenvalue problem involving an ordinary differential equation for the function f special attention is paid to the case when m n 2 over n 2 then beta 0 and the function f can be found explicity the proof is based on a geometric lyapunov type argument and comparison arguments based on the intersection properties of the solution graphs

perturbation results of critical elliptic equations of caffarelli kohn nirenberg type abstract we find for small e positive solutions to the equation div x 2a u x 2 1 a u 1 ek x u p 1 x bp in r n which branch off from the manifold of minimizers in the class of radial functions of the corresponding caffarelli kohn nirenberg type inequality moreover our analysis highlights the symmetry breaking phenomenon in these inequalities namely the existence of non radial minimizers

sobolev and hardy littlewood sobolev inequalities this paper is devoted to improvements of sobolev and onofri inequalities the additional terms involve the dual counterparts i e hardy littlewood sobolev type inequalities we focus our attention on optimal constants that can be achieved either by completion of the square methods or by using nonlinear flows and provide various new estimates

granular matrix a new approach for granular structure reduction and redundancy evaluation granular structure is a mathematical expression of knowledge in granular computing and a direct determinant of the data processing efficiency to improve the efficiency of data processing many scholars have studied the reduction of granular structure the attribute reduction and the granular reduction are two types of reduction on different layers of a granular structure with the latter being both an essential step for granular structure reduction and the foundation of the attribute reduction yet compared with the attribute reduction the granular reduction has received less attention from scholars therefore a fuzzy granular reduction theory and a granular matrix based on the fuzzy beta coverings is proposed in this article the insufficiency of the existing granular reduction theory for fuzzy beta coverings is pointed out and proper sufficient and necessary conditions for two fuzzy beta coverings generating the same upper and lower approximations are also given in this article in addition to reduce and evaluate a fuzzy beta covering a novel reduction algorithm based on a granular matrix is proposed for the first time also since fuzzy covering reduction is np hard a heuristic greedy algorithm is designed to obtain a reduct numerical experiments show that the redundancy rates of neighborhood granule sets induced by some big scale data sets exceed 99 which indicates that the existing neighborhood granulation methods need to be urgently improved based on this concise granular structures and much more efficient feature selection algorithms can be proposed in the future

new results on single axioms for l fuzzy rough approximation operators abstract this paper further studies axiomatic characterizations of l fuzzy rough sets where l denotes a residuated lattice single axioms for upper l fuzzy rough approximation operators are investigated in two different approaches which describe upper l fuzzy approximation operators with ordinary l fuzzy operations and l fuzzy product operation respectively we characterize upper l fuzzy rough approximation operators by only one axiom with ordinary l fuzzy operations when the l fuzzy relation is euclidean as well as the composition of euclidean with serial resp reflexive we also discuss the single axioms for upper l fuzzy rough approximation operators with l fuzzy product operation on arbitrary l fuzzy relations that are not limited as either general l fuzzy relations or symmetric l fuzzy relations the single axioms for lower l fuzzy rough approximation operators are discussed with ordinary l fuzzy operations regardless of the regularity of residuated lattice l

on fuzzy rough sets based on tolerance relations dubois and prade 1990 1 introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets which was originally proposed by pawlak 1982 8 later radzikowska and kerre introduced the so called i t fuzzy rough sets where i is an implication and t is a triangular norm in the present paper by using a pair of implications i j we define the so called i j fuzzy rough sets which generalize the concept of fuzzy rough sets in the sense of radzikowska and kerre and that of mi and zhang basic properties of i j fuzzy rough sets are investigated in detail

generalized rough sets based on reflexive and transitive relations in this paper we investigate the relationship between generalized rough sets induced by reflexive and transitive relations and the topologies on the universe which is not restricted to be finite it is proved that there exists a one to one correspondence between the set of all reflexive and transitive relations and the set of all topologies which satisfy a certain kind of compactness condition

neighborhood operators for covering based rough sets covering based rough sets are important generalizations of the classical rough sets of pawlak a common way to shape lower and upper approximations within this framework is by means of a neighborhood operator in this article we study 24 such neighborhood operators that can be derived from a single covering we verify equalities between them reducing the original collection to 13 different neighborhood operators for the latter we establish a partial order showing which operators yield smaller or greater neighborhoods than others six of the considered neighborhood operators result in new covering based rough set approximation operators we study how these new approximation operators relate to existing ones in terms of partial order relations i e whether the generated approximations are in general greater smaller or incomparable finally we discuss the connection between the covering based approximation operators and relation based approximation operators another prominent generalization of pawlak s rough sets

localized upper bounds of heat kernels for diffusions via a multiple dynkin hunt formula we prove that for a general diffusion process certain assumptions on its behavior only within a fixed open subset of the state space imply the existence and sub gaussian type off diagonal upper bounds of the global heat kernel on the fixed open set the proof is mostly probabilistic and is based on a seemingly new formula which we call a multiple dynkin hunt formula expressing the transition function of a hunt process in terms of that of the part process on a given open subset this result has an application to heat kernel analysis for the liouville brownian motion the canonical diffusion in a certain random geometry of the plane induced by a massive gaussian free field

an extended gentzen type formulation of a many valued modal propositinal logic based on zadeh s similarity relation the main purpose of this paper is to present an extended gentzen type formulation of a many valued modal propositional logic based on zadeh s similarity relation

rough set models in multigranulation spaces there exist several approaches to rough set approximations in a multigranulation space namely a family of equivalence relations in this paper we propose a unified framework to classify and compare existing studies an underlying principle is to explain rough sets in a multigranulation space through rough sets derived by using individual equivalence relations two basic models are suggested one model is based on a combination of a family of equivalence relations into an equivalence relation and the construction of approximations with respect to the combined relation by combining equivalence relations through set intersection and union respectively we construct two sub models the other model is based on the construction of a family of approximations from a set of equivalence relations and a combination of the family of approximations by using set intersection and union to combine a family of approximations respectively we again build two sub models as a result we have a total of four models we examine these models and give conditions under which some of them become the same

a distance exponent for liouville quantum gravity let gamma in 0 2 and let h be the random distribution on mathbb c which describes a gamma liouville quantum gravity lqg cone also let kappa 16 gamma 2 4 and let eta be a whole plane space filling sle kappa curve sampled independent from h and parametrized by gamma quantum mass with respect to h we study a family mathcal g epsilon epsilon 0 of planar maps associated with h eta called the lqg structure graphs a k a mated crt maps which we conjecture converge in probability in the scaling limit with respect to the gromov hausdorff topology to a random metric space associated with gamma lqg in particular mathcal g epsilon is the graph whose vertex set is epsilon mathbb z with two such vertices x 1 x 2 in epsilon mathbb z connected by an edge if and only if the corresponding curve segments eta x 1 epsilon x 1 and eta x 2 epsilon x 2 share a non trivial boundary arc due to the peanosphere description of sle decorated lqg due to duplantier et al liouville quantum gravity as a mating of trees 2014 the graph mathcal g epsilon can equivalently be expressed as an explicit functional of a correlated two dimensional brownian motion so can be studied without any reference to sle or lqg we prove non trivial upper and lower bounds for the cardinality of a graph distance ball of radius n in mathcal g epsilon which are consistent with the prediction of watabiki prog theor phys suppl 114 1 17 1993 for the hausdorff dimension of lqg using subadditivity arguments we also prove that there is an exponent chi 0 for which the expected graph distance between generic points in the subgraph of mathcal g epsilon corresponding to the segment eta 0 1 is of order epsilon chi o epsilon 1 and this distance is extremely unlikely to be larger than epsilon chi o epsilon 1

a note on dynamic relational systems abstract it is known that weak and strong definabilities are defined for multiple source approximation systems by khan and banerjee this paper presents in a more general setting a discussion on definabilities of sets in dynamic relational systems we prove that the inverse image of a weak or strong definable set with respect to relation preserving function is also weak or strong definable set respectively on the way we show that the inverse image of a reduct of attribute set is also a reduct under the object function of an information system homomorphism further we give the connections between definable sets and the topology determined by the intersections of the topologies of reflexive relations a quasi uniformity is a filter on the cartesian product of a given universe satisfying certain conditions in fact every quasi uniformity is a dynamic relational system where the relations are reflexive in this respect we discuss on the connections between approximation systems and quasi uniformities

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

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

an application of covering approximation spaces on network security based on the pawlak rough set theory this paper investigates separations in covering approximation spaces give some characterizations of these separations and some relations among these separations as an application of these results investigations on network security are converted into investigations on separations in covering approximation spaces by taking covering approximation spaces as mathematical models of networks results of this paper give further applications of the pawlak rough set theory in pattern recognition and artificial intelligence which makes it possible to research network security by logical methods and mathematical methods in computer science this contributes to giving risk assessments of securities and to raise grades of securities for networks

gaussian multiplicative chaos revisited in this article we extend the theory of multiplicative chaos for positive definite functions in rd of the form f x 2 ln t x g x where g is a continuous and bounded function the construction is simpler and more general than the one defined by kahane in 1985 as main application we give a rigorous mathematical meaning to the kolmogorov obukhov model of energy dissipation in a turbulent flow

a study of rough sets based on 1 neighborhood systems abstract a special class of neighborhood systems called 1 neighborhood systems are important in rough set theory by using a concept core originated in general topology we define two types of rough sets based on 1 neighborhood systems in this paper we discuss properties of these rough sets from the perspective of both common 1 neighborhood systems and several special classes of 1 neighborhood systems such as reflexive symmetric transitive or euclidean 1 neighborhood systems by using these properties we discuss the relationship among several classes of 1 neighborhood systems with various special properties we give a necessary and sufficient condition for a reflexive and symmetric 1 neighborhood system being a unary we also prove that a reflexive and transitive 1 neighborhood system is representative the proofs of these results show that the rough sets we defined in this paper not only have application background but also have theoretic importance

topological properties of generalized approximation spaces rough set theory is a powerful mathematical tool for dealing with inexact uncertain or vague information the core concepts of rough set theory are information systems and approximation operators of approximation spaces approximation operators draw close links between rough set theory and topology this paper concerns generalized approximation spaces via topological methods and studies topological properties of rough sets classical separation axioms compactness and connectedness for topological spaces are extended to generalized approximation spaces relationships among separation axioms for generalized approximation spaces and relationships between topological spaces and their induced generalized approximation spaces are investigated an example is given to illustrate a new approach to recover missing values for incomplete information systems by regularity of generalized approximation spaces

on a ktb modal fuzzy logic abstract an extension of the classical ktb modal logic to the fuzzy case is considered a new notion of similarity is introduced and a fuzzy logic based on this relation is proposed also the validity of formulas is defined by introducing a notion of logical consequence then the decision procedure of this logic is described in the tableau method

l approximation spaces and l fuzzy quasi uniform spaces we introduce the notion of generalized l upper and generalized l lower approximation spaces under a family of information systems we define l upper and l lower quasi approximation operators in fuzzy rough sets and l fuzzy quasi uniform spaces as a view point of the entourage approach in a strictly two sided commutative bi quantale we investigate the relations between l quasi approximation spaces and l fuzzy quasi uniform spaces

approximation operators on complete completely distributive lattices rough set a tool for data mining deals with the vagueness and granularity in information systems in 2006 chen et al initiated the study of rough approximations on a complete completely distributive lattice ccd lattice for short and brought generalizations of rough sets into a unified framework in this paper we discuss the approximation operators on a ccd lattice based on the concept of neighborhood three kinds of upper approximation operators and a kind of lower approximation operator are constructed basic properties of lower and upper approximation operators are examined furthermore the relationships among these operators are analyzed

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

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

fuzzy reasoning based on a new fuzzy rough set and its application to scheduling problems in this paper we study the fuzzy reasoning based on a new fuzzy rough set first we define a broad family of new lower and upper approximation operators of fuzzy sets between different universes using a set of axioms then based on the approximation operators above we propose the fuzzy reasoning based on the new fuzzy rough set by means of the above fuzzy reasoning based on the new fuzzy rough set for a given premise we can obtain the fuzzy reasoning consequence expressed by the fuzzy interval constructed by the above two approximations of fuzzy sets furthermore through the defuzzification of the lower and upper approximations we can get the corresponding two values constructing the interval used as the fuzzy reasoning consequence after defuzzification then from the above interval a suitable value can be selected as the final reasoning consequence so that some special constraints are satisfied as possibly at last we apply the fuzzy reasoning based on the new fuzzy rough set to the scheduling problems and numerical computational results show that the fuzzy reasoning based on the new fuzzy rough set is more suitable for the scheduling problems compared with the fuzzy reasoning based on the cri method and the iii method

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

using one axiom to characterize l fuzzy rough approximation operators based on residuated lattices abstract axiomatic characterization of approximation operators plays an important role in the study of rough set theory different axiom sets of abstract operators can illustrate different classes of rough set systems in this paper we are devoted to searching for a single axiom to characterize l fuzzy rough approximation operators based on residuated lattices axioms of l fuzzy set theoretic operators make sure of the existence of certain types of l fuzzy relations which produce the same operators we demonstrate that the lower upper l fuzzy rough approximation operators generated by a generalized l fuzzy relation can be characterized by only one axiom furthermore we also use one axiom to characterize l fuzzy rough approximation operators produced by the l fuzzy serial reflexive symmetric and t transitive relations as well as any of their compositions

three types of fuzzy covering based rough set models abstract in this paper three new types of fuzzy covering based rough set models are proposed first the membership degree based fuzzy covering rough set is defined in the fuzzy covering approximation space by separately comparing the membership degree of each object belonging to the approximated fuzzy set with those of the component fuzzy sets in the fuzzy covering for the same object second the membership function based fuzzy covering rough set is obtained by globally comparing the membership function of the approximated fuzzy set with those of the component fuzzy sets in the fuzzy covering finally the membership function based fuzzy covering rough set model is generalized to the fuzzy covering approximation space the properties of these new rough set models are discussed furthermore the relationship between these new rough sets and the relationship between the new rough set and pawlak rough set are described

rough set analysis of a general type of fuzzy data using transitive aggregations of fuzzy similarity relations abstract in this paper we initially present and compile the tools and theories that are necessary for a rough set analysis of a general type of fuzzy data from this study we have developed two original contributions that can enhance the application of fuzzy rough sets techniques to data that may be affected by several sources of uncertainty in its measurement process for this type of analysis we have first considered the construction of homogeneous granules in which reflexivity symmetry and a certain transitivity property are satisfied this has encouraged us to study the application of t similarity relations in this framework and to examine the available theorems that permit us to aggregate this type of relations as a part of this study we have also developed a new theorem that allows us to use the dual of the generalized means as an aggregation operator in the fuzzy rough sets context another theoretical development that tries to respond to a practical problem is the concept of precision aggregation we use this new notion to export ziarko s variable precision rough set model to the analysis of fuzzy data a generalization that becomes a necessity for the fuzzy rough sets analysis of large databases the utility of this approach is demonstrated with an application example in the vehicle database donated by the turing institute

on lower and upper intension order relations by different cover concepts in this paper the concept of intension is used to introduce two types of ordering relations based on information that generates a cover for the universal set these types of ordering relations are distinct from the well known ordering relation based on set inclusion for these ordering relations we consider the algebraic structures that arise in various types of covers we show that in the case of a representative cover the algebraic structure resulting from the lower intension inclusion is a double stone algebra while in the case of a reduced cover it is a boolean algebra in addition the algebraic structure resulting from the upper intension inclusion in the case of a representative cover is a boolean algebra and in the case of a reduced cover the two boolean algebraic structures from lower and upper intension inclusions are isomorphic

fuzzy rough set theory for the interval valued fuzzy information systems the concept of the rough set was originally proposed by pawlak as a formal tool for modelling and processing incomplete information in information systems then in 1990 dubois and prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets the aim of this paper is to present a new extension of the rough set theory by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory i e the interval valued fuzzy rough set model is presented based on the interval valued fuzzy information systems which is defined in this paper by a binary interval valued fuzzy relations r f i uxu on the universe u several properties of the rough set model are given and the relationships of this model and the others rough set models are also examined furthermore we also discuss the knowledge reduction of the classical pawlak information systems and the interval valued fuzzy information systems respectively finally the knowledge reduction theorems of the interval valued fuzzy information systems are built

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

a fuzzy covering based rough set model and its generalization over fuzzy lattice in this paper we study a new type of fuzzy covering based rough set model by introducing the notion of fuzzy minimal description we mainly address the following issues in this paper first we present the definition of fuzzy minimal description and study its properties then we define a novel type of fuzzy covering based rough set model and investigate the properties of this model furthermore the axiomizations and matrix representations of the fuzzy covering lower and fuzzy covering upper approximations are the vital problems we investigate in this paper moreover we explore the conditions under which two fuzzy coverings generate the same fuzzy covering lower or fuzzy covering upper approximations finally we generalize the model to l fuzzy covering based rough set which is defined over fuzzy lattices similar to the fuzzy covering based rough set model we also address the issues mentioned above to the l fuzzy covering based rough set

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

continuity and estimates of the liouville heat kernel with applications to spectral dimensions the liouville brownian motion lbm recently introduced by garban rhodes and vargas and in a weaker form also by berestycki is a diffusion process evolving in a planar random geometry induced by the liouville measure m gamma formally written as m gamma dz e gamma x z gamma 2 mathbb e x z 2 2 dz gamma in 0 2 for a massive gaussian free field x it is an m gamma symmetric diffusion defined as the time change of the two dimensional brownian motion by the positive continuous additive functional with revuz measure m gamma in this paper we provide a detailed analysis of the heat kernel p t x y of the lbm specifically we prove its joint continuity a locally uniform sub gaussian upper bound of the form p t x y le c 1 t 1 log t 1 exp bigl c 2 x y beta wedge 1 t frac 1 beta 1 bigr for t in bigl 0 frac 1 2 bigr for each beta frac 1 2 gamma 2 2 and an on diagonal lower bound of the form p t x x ge c 3 t 1 bigl log t 1 bigr eta for t in 0 t eta x with t eta x in bigl 0 frac 1 2 bigr heavily dependent on x for each eta 18 for m gamma almost every x as applications we deduce that the pointwise spectral dimension equals 2 m gamma a e and that the global spectral dimension is also 2

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

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

symmetry of extremals of functional inequalities via spectral estimates for linear operators we prove new symmetry results for the extremals of the caffarelli kohn nirenberg inequalities in any dimension larger or equal than 2 in a range of parameters for which no explicit results of symmetry were previously known

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

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 logic for fuzzy data analysis abstract the aim of this paper is to present new ideas on how to analyse fuzzy data using logic tools we propose a modal logic based on fuzzy relations special attention is paid to similarity relations on a set of objects we discuss its relationships to rough sets then we give useful properties for transformation of formulas

estimates for the optimal constants in multipolar hardy inequalities for schrodinger and dirac operators by expanding squares we prove several hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities these inequalities involve the l2 norm such results are generalized to an arbitrary number of singularities and compared with standard results given by the ims method the generalized version of hardy inequalities with several singularities is equivalent to some spectral information on a schrodinger operator involving a potential with several inverse square singularities we also give a generalized hardy inequality for dirac operators in the case of a potential having several singularities of coulomb type which are critical for dirac operators

communication between information systems using fuzzy rough sets communication between information systems is a basic problem in granular computing and the concept of homomorphism is a useful mathematical tool to study this problem in this paper some properties of communication between information systems based on fuzzy rough sets are investigated the concepts of fuzzy relation mappings between universes are first proposed in order to construct a fuzzy relation of one universe according to the given fuzzy relation on the other universe the main properties of the mappings are studied the notions of homomorphism of information systems based on fuzzy rough sets are then proposed and it is proved that properties of relation operations in the original information system and structural features of the system such as approximations of arbitrary fuzzy sets and attribute reductions are guaranteed in its image system under the condition of homomorphism

the necessary and sufficient conditions for a fuzzy relation being τ euclidean the connections between special types of fuzzy relations and properties of fuzzy rough approximation operators have been established in recent years but euclidean fuzzy relation has not been considered yet in the present paper the necessary and sufficient conditions for a fuzzy relation being euclidean are investigated in three different fuzzy rough approximation spaces meanwhile it is proved that in i fuzzy rough approximation space where i is an r implication the properties the euclidean i fuzzy rough approximation operators possess are just the same as those in rough fuzzy approximation space

on definable concepts of rough set models in various generalized rough set models definable concepts are used to approximate other concepts based on a unified abstract rough set theory this paper investigates the mathematical structure of the set of definable concepts in several generalized rough set models such as relation based models covering based models and fuzzy based models it is shown that there exist two kinds of interesting structures for the set of definable concepts which are often used in rough set models the complementary structure and the equational structure in addition relations between different rough set models are discussed

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

the relationship between l fuzzy rough set and l topology various fuzzy generalizations of rough approximations have been proposed in the literature this paper is devoted to the discussion of the relationship between l fuzzy rough sets and l topologies on an arbitrary universe finally one to one correspondence between the set of all reflexive transitive l relations and the set of all alexandrov l topologies is obtained

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

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models