rough prime ideals and rough fuzzy prime ideals in semigroups in this paper we shall introduce the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup and give some properties of such ideals also we will discuss the relations between the upper lower rough prime ideals and the upper lower approximations of their homomorphism images

using one axiom to characterize rough set and fuzzy rough set approximations axiomatic approaches are important for understanding the concepts of rough set theory the properties of the approximation operators constructed in rough set theory are determined by a binary relation a linkage in the form of a necessary and sufficient condition between the constructive approach and an axiomatic system is established in this paper various classes of rough set algebras are characterized by different sets of axioms with no restriction on the cardinality of the universal set we use only one axiom to describe the approximation generated by the reflexive symmetric and transitive relations respectively in addition we characterize lower and upper approximations of the pawlak rough set with only one axiom we also study a similar problem in the context of fuzzy sets

ergodic properties of color records abstract in a series of papers culminating in the dissertation of the second author under the direction of p w kasteleyn several rigorous results have been obtained concerning the color sequence seen by a random walker on a stochastically black white colored lattice and this under the sole assumption that the coloring is stationary ergodic and independent of the walking in this article we investigate the ergodic properties of the color record process and their implications for the behavior of the sequence nk k 0 of successive times needed for the walker to go from the kth to the k 1 st black point as an example of the type of results obtained we show that if a certain induced color record process is strongly mixing in the ergodic theoretic sense then the average nk tends to the inverse of the density of black points as k tends to infinity and similarly that the distribution of nk tends to a limit that can be explicitly calculated in terms of the distribution of n0 for a certain class of stochastic colorings and random walks we are able to verify that the induced color record process is indeed strongly mixing our main goal however is to clarify the nature of the ergodic properties and thereby provide a tool for future use in the study of color records

strong invariance for local times let y 1 y 2 be a sequence of i i d random variables with distribution p y 1 k p k k 1 2 e y 1 0 e y 1 2 2 put t n y 1 y n and n x n k 0 k n t k x extending the result of revesz 1981 it is shown that for appropriate skorohod construction we have 1 mathop text sup limits x in mathbb z text l x n sigma 2 text sigma text 2 n x n o n 1 mathord left vphantom 1 4 varepsilon right kern nulldelimiterspace 4 varepsilon text a text s text provided all moments e y 1 m m 0 exists where l is the local time of a wiener process certain rate of convergence is given also under weaker conditions and for l x n 2 2 n x n too when x is fixed

variations on a theme by mark kac mark kac s theorem on the mean recurrence time in a stationary stochastic process in discrete time with discrete states is taken as the starting point for a series of variations most of which are formulated in terms of 0 1 processes whereas the original theorem deals with the mean recurrence time of a given state under the condition that the state is realized at time 0 this condition is dropped in part of the variations two others refer to the variance of the recurrence time and two to the poincare cycle of a dynamical system most variations consist in inequalities and formal identities for the mean first arrival time and subsequent recurrence times for the given state

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

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

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

dominance based rough fuzzy set approach and its application to rule induction abstract theories of fuzzy sets and rough sets are related and complementary methodologies to handle uncertainty of vagueness and coarseness respectively marrying both leads to the hybrid notion of rough fuzzy sets in order to get a more accurate account of imperfect information in this paper our attention is paid to ordered fuzzy decision systems where condition criteria are preference ordered and decision classes are not only ordered but also fuzzy first the dominance based rough fuzzy approximations of an upward or downward cumulated fuzzy set are introduced in ordered fuzzy decision systems second lower and upper reducts relative to a certain cumulated fuzzy set are proposed to eliminate redundant criteria in the system then two approaches to attribute reduction are presented based on the discernibility matrix and the heuristic strategy respectively also decision rules are extracted directly from these approximations and some applicable and simplified decision rules are obtained according to requirements of decision makers finally a case study in bankruptcy risk analysis is used to illustrate the mechanism of the proposed methods

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

analytical and numerical comparisons of biogeography based optimization and genetic algorithms we show that biogeography based optimization bbo is a generalization of a genetic algorithm with global uniform recombination ga gur based on the common features of bbo and ga gur we use a previously derived bbo markov model to obtain a ga gur markov model one bbo characteristic which makes it distinctive from ga gur is its migration mechanism which affects selection pressure i e the probability of retaining certain features in the population from one generation to the next we compare the bbo and ga gur algorithms using results from analytical markov models and continuous optimization benchmark problems we show that the unique selection pressure provided by bbo generally results in better optimization results for a set of standard benchmark problems we also present comparisons between bbo and ga gur for combinatorial optimization problems include the traveling salesman the graph coloring and the bin packing problems

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

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

on the generalized multinomial distribution optimal multinomial detectors and generalized weighted partial decision detectors the probability of a particular outcome of a multinomial trial may depend on the trial number the generalized multinomial distribution describes such an experiment a mathematical statement of the generalized multinomial distribution is given the neyman pearson optimum generalized multinomial detectors for deterministic signals are derived for both binary and m ary signaling the equivalence of the generalized multinomial detectors and generalized weighted partial decision detectors is shown the author claims that the work of e k al hussaini see ibid vol 37 p 1099 1989 on generalized multinomial detectors contains certain errors

tail triviality for sums of stationary random variables we study tail sigma fields and loss of memory associated with sums of stationary integer valued random variables an application concerns convergence in distribution of interarrival times in zero one sequences

rough approximation of a fuzzy concept on a hybrid attribute information system and its uncertainty measure abstract one important and valuable topic in rough set theory is the concept of approximation over various generalized information systems although rough set models and approaches over hybrid attribute sets have been studied by many researchers recently the studies focus on the rough approximation of a crisp concept over a hybrid attribute information system this paper considers the rough approximation of a fuzzy concept over a hybrid attribute information system we define a hybrid indiscernibility relation by compounding the fuzzy indiscernibility similarity relation over numerical attributes with the equivalence relation over symbolic attributes then we present the lower and upper approximations of a fuzzy set based on the hybrid indiscernibility relation i e the fuzzy rough set over a hybrid attribute information system also some interesting properties of the hybrid fuzzy rough approximation operators are presented in detail and the relationship between the fuzzy rough set and the existing rough set model over a hybrid attribute information system is established meanwhile based on the definition of the cut set of a fuzzy set of the universe of discourse the representations of the lower and upper approximations of a fuzzy set are given with respect to hybrid attribute approximation space furthermore the concept of the cut set of the hybrid indiscernibility relation of the hybrid attribute approximation space is defined then the characterizations of the hybrid fuzzy rough approximation operators are established as well two different types of characterization theorems of the hybrid fuzzy rough approximation operators are presented based on the cut set of the hybrid indiscernibility relation over the hybrid attribute information system at the same time we study the uncertainty measure of the fuzzy rough set over a hybrid attribute information system by using knowledge granulation the main contribution of this paper is twofold one is to extend the existing rough set approach over hybrid attribute information systems to a fuzzy environment another is to present a new way to define the rough approximation of a fuzzy concept as well the uncertainty measure over hybrid attribute information systems

mixing properties for random walk in random scenery consider the lattice z d d geq 1 together with a stochastic black white coloring of its points and on it a random walk that is independent of the coloring a local scenery perceived at a given time is a pattern of colors seen by the walker in a finite box around his current position under weak assumptions on the probability distributions governing walk and coloring we prove asymptotic independence of local sceneries perceived at times 0 and n in the limit as n rightarrow infty and at times 0 and t k in the limit as k rightarrow infty where t k is the random k th hitting time of a black point an immediate corollary of the latter result is the convergence in distribution of the interarrival times between successive black hits i e of t k 1 t k as k rightarrow infty the limit distribution is expressed in terms of the distribution of the first hitting time t 1 the proof uses coupling arguments and ergodic theory

reasoning within expressive fuzzy rough description logics it is generally accepted that the management of imprecision and vagueness will yield more intelligent and realistic knowledge based applications description logics dls are suitable well known logics for managing structured knowledge that have gained considerable attention the last decade the current research progress and the existing problems of uncertain or imprecise knowledge representation and reasoning in dls are analyzed in this paper an integration between the theories of fuzzy dls and rough dls has been attempted by providing fuzzy rough dls based on fuzzy rough set theory the syntax semantics and properties of fuzzy rough dls are given it is proved that the satisfiability subsumption entailment and abox consistency reasoning in fuzzy rough dls may be reduced to the abox consistency reasoning in the corresponding fuzzy dls

granulation of a fuzzy set nonspecificity granulation decomposition of a fuzzy set a defined on a finite set of objects x is studied two types of decomposition are considered external granulation determined by a given equivalence relation on x and internal granulation created by clusters of elements from x with similar membership grades in a axiomatic definitions of measures of granular nonspecificity and granular specificity are proposed some general approaches to the construction of measures of granular nonspecificity specificity are suggested relationship between granular nonspecificity roughness and nonspecificity of a fuzzy set is discussed

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

reconstructing a random scenery observed with random errors along a random walk path we show that an i i d uniformly colored scenery on observed along a random walk path with bounded jumps can still be reconstructed if there are some errors in the observations we assume the random walk is recurrent and can reach every point with positive probability at time k the random walker observes the color at her present location with probability 1 and an error y k with probability the errors y k k 0 are assumed to be stationary and ergodic and independent of scenery and random walk if the number of colors is strictly larger than the number of possible jumps for the random walk and is sufficiently small then almost all sceneries can be almost surely reconstructed up to translations and reflections

roughness bounds in rough set operations this paper presents some roughness bounds for rough set operations the results show that a bound of the set operation can be determined from their operand s roughnesses we prove also that this bound is not determined under some special operations

retrieving random media benjamini asked whether the scenery reconstruction methods of matzinger see e g 21 22 20 can be done in polynomial time in this article we give the following answer for a 2 color scenery and simple random walk with holding we prove that a piece of the scenery of length of the order 3 n around the origin can be reconstructed up to a reflection and a small translation with high probability from the first 2 310 n observations with a constant 0 independent of n thus the number of observations needed is polynomial in the length of the piece of scenery which we reconstruct the probability that the reconstruction fails tends to 0 as n

a new methodology for fuzzification of rough sets based on α indiscernibility abstract this paper presents a new framework for fuzzification of rough sets when the objects are indiscernible that is indiscernible up to a certain degree to this end a fuzzy proximity relation has been used basic properties of the so called fuzzified rough sets are examined following radzikowska and kerre 28 approach the technique is further generalized using t norms and implication operators two algorithms followed by an example have also been presented for practical applications of the new technique

weighted fuzzy interpolative reasoning for sparse fuzzy rule based systems based on piecewise fuzzy entropies of fuzzy sets in this paper we propose a new method for weighted fuzzy interpolative reasoning in sparse fuzzy rule based systems based on piecewise fuzzy entropies of fuzzy sets first the proposed method uses the representative values of antecedent fuzzy sets the representative values of observation fuzzy sets and the representative values of consequence fuzzy sets of fuzzy rules to get the characteristic points of the fuzzy interpolative result represented by a fuzzy set then it calculates the piecewise fuzzy entropies between any two characteristic points of the antecedent fuzzy sets the piecewise fuzzy entropies between any two characteristic points of the observation fuzzy sets and the piecewise fuzzy entropies between any two characteristic points of the consequence fuzzy sets of the fuzzy rules respectively then it calculates the weights of the antecedent fuzzy sets of each fuzzy rule respectively and calculates the weight of each fuzzy rule then it calculates the piecewise fuzzy entropies between any two characteristic points of the fuzzy interpolative result finally it uses the secant method to calculate the degree of membership of each obtained characteristic point of the fuzzy interpolative result the experimental results show that the proposed method outperforms the existing methods for dealing with the multivariate regression problems the mackey glass chaotic time series prediction problem and the time series prediction problems

measuring roughness of generalized rough sets induced by a covering in this paper we propose new lower and upper approximations and obtain some important properties in generalized rough set induced by a covering especially these properties are compared with ones of pawlak s rough sets and bonikowski s covering generalized rough sets respectively moreover we define a measure of roughness based on generalized rough sets with the new approximations and discuss some significant properties of the measure

the algebraic structures of generalized rough set theory rough set theory is an important technique for knowledge discovery in databases and its algebraic structure is part of the foundation of rough set theory in this paper we present the structures of the lower and upper approximations based on arbitrary binary relations some existing results concerning the interpretation of belief functions in rough set backgrounds are also extended based on the concepts of definable sets in rough set theory two important boolean subalgebras in the generalized rough sets are investigated an algorithm to compute atoms for these two boolean algebras is presented

fuzzy rough granular neural networks fuzzy granules and classification we introduce a fuzzy rough granular neural network frgnn model based on the multilayer perceptron using a back propagation algorithm for the fuzzy classification of patterns we provide the development strategy of the network mainly based upon the input vector initial connection weights determined by fuzzy rough set theoretic concepts and the target vector while the input vector is described in terms of fuzzy granules the target vector is defined in terms of fuzzy class membership values and zeros crude domain knowledge about the initial data is represented in the form of a decision table which is divided into subtables corresponding to different classes the data in each decision table is converted into granular form the syntax of these decision tables automatically determines the appropriate number of hidden nodes while the dependency factors from all the decision tables are used as initial weights the dependency factor of each attribute and the average degree of the dependency factor of all the attributes with respect to decision classes are considered as initial connection weights between the nodes of the input layer and the hidden layer and the hidden layer and the output layer respectively the effectiveness of the proposed frgnn is demonstrated on several real life data sets

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

fuzzy entropy a more comprehensible perspective for interval shadowed sets of fuzzy sets shadowed sets proposed by pedrycz provide a three way approximation scheme for transforming the universe of a fuzzy set into three disjoint areas i e elevated area reduced area and shadow area to calculate a pair of decision making thresholds an analytic method was proposed by solving a minimization problem of the uncertainty arising from the three areas however some uncertainty will be lost in the process of constructing the shadowed set model using pedrycz s method moreover few references on how to measure the uncertainty of shadow sets exist in this article a comprehensible method for measuring the fuzzy entropy of a shadowed set i e interval fuzzy entropy is defined based on the interval fuzzy entropy a new shadowed set model namely interval shadowed sets is proposed compared with pedrycz s model the main difference is that the range of the shadow area in this model is beta alpha 0 leq beta while not 0 1 by solving a fuzzy entropy loss minimization problem a pair of optimal thresholds alpha and beta can be obtained finally the results of the instance analysis of different types of representative membership functions and many experiments show that the fuzzy entropy loss of the interval shadowed set is lower than that of the traditional shadowed set of a fuzzy set these results enrich shadowed set theory from a new perspective

rough fuzzy rule interpolation abstract fuzzy rule interpolation forms an important approach for performing inference with systems comprising sparse rule bases even when a given observation has no overlap with the antecedent values of any existing rules fuzzy rule interpolation may still derive a useful conclusion unfortunately very little of the existing work on fuzzy rule interpolation can conjunctively handle more than one form of uncertainty in the rules or observations in particular the difficulty in defining the required precise valued membership functions for the fuzzy sets that are used in conventional fuzzy rule interpolation techniques significantly restricts their application in this paper a novel rough fuzzy approach is proposed in an attempt to address such difficulties the proposed approach allows the representation handling and utilisation of different levels of uncertainty in knowledge this allows transformation based fuzzy rule interpolation techniques to model and harness additional uncertain information in order to implement an effective fuzzy interpolative reasoning system final conclusions are derived by performing rough fuzzy interpolation over this representation the effectiveness of the approach is illustrated by a practical application to the prediction of diarrhoeal disease rates in remote villages it is further evaluated against a range of other benchmark case studies the experimental results confirm the efficacy of the proposed work

decision theoretic three way approximations of fuzzy sets abstract a three way three valued or three region approximation of a fuzzy set is constructed from a pair of thresholds on the fuzzy membership function an element whose membership grade equals to or is greater than is put into the positive region an element whose membership grade equals to or is less than is put into the negative region and an element whose membership grade is between and is put into the boundary region a fundamental issue is the determination and interpretation of the required pair of thresholds in the framework of shadowed sets i e an example of three way approximations of fuzzy sets pedrycz provides an analytic solution to computing the thresholds by searching for a balance of uncertainty introduced by the three regions to gain further insights into three way approximations of fuzzy sets we introduce an alternative decision theoretic formulation in which the required thresholds are computed by minimizing decision cost

fuzzy interpolative reasoning based on the ratio of fuzziness of rough fuzzy sets in this paper we propose a method to construct a polygonal rough fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts opinions and propose a new fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on the ratio of fuzziness of the constructed polygonal rough fuzzy sets where the values of the antecedent variables and the consequence variable appearing in the fuzzy rules are represented by the constructed polygonal rough fuzzy sets the proposed fuzzy interpolative reasoning method can overcome the drawbacks of the existing method due to the fact that it can deal with fuzzy interpolative reasoning using polygonal rough fuzzy sets and it gets more reasonable fuzzy interpolative reasoning results than the existing method

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

the notion of roughness of a fuzzy set abstract the roughness of a set according to the notion introduced by pawlak in 1991 can be regarded as the mz distance between its upper and the lower approximations with this idea in mind we have generalized pawlak s definition by replacing the mz distance by a general distance measure we also generalize the notion of roughness of fuzzy sets introduced by huyhn and nakamori in 2005

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

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

can fuzzy entropies be effective measures for evaluating the roughness of a rough set the roughness of a rough set arises from the existence of its boundary region in such a boundary region each object has a non zero rough membership degree when an object s rough membership degree is regarded as its fuzzy membership degree a rough set can induce a fuzzy set this relationship motivates us to assert that there may exist some inherent relations between the roughness of a rough set and the fuzziness of the fuzzy set induced from the rough set this assertion leads us to the question can the existing fuzzy entropies be used to evaluate the roughness of a rough set to answer this question we first analyze how the boundary region varies when the partition of the universe becomes coarser and then exploit this analysis in the introduction of a more appropriate definition on the roughness of a rough set to determine whether a fuzzy entropy can be used to evaluate the roughness of a rough set or not we develop three methods for estimating the ability of a fuzzy entropy to measure the roughness the experiments show that these methods are very effective and can be applied to select a fuzzy entropy as a measure of the roughness of a rough set

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

a long time tail for random walk in random scenery consider a simple random walk on d whose sites are colored black or white independently with probabilityq resp 1 q walk and coloring are independent letn k be the number of steps by the walk between itskth and k 1 th visits to a black site i e the length of itskth white run and let k e n k q 1 our main result is a proof that lim k k d 2 k 1 q q d 2 2 d 2 d 2 since it is known thatq 1 k e n 1 n k 1 b e n 1 b e n k 1 b withb the event that the origin is black exhibits a long time tail in the run length autocorrelation function numerical calculations of k 1 k 100 ind 1 2 and 3 show that there is an oscillatory behavior of k for smallk this damps exponentially fast following which the power law sets in fairly rapidly we prove that if the coloring is not independent but is convex in the sense of fkg then the decay of k cannot be faster than

a roughness measure for fuzzy sets recently an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets m banerjee s k pal roughness of a fuzzy set information sciences 93 1996 235 246 essentially banerjee and pal s roughness measure depends on parameters that are designed as thresholds of definiteness and possibility in membership of the objects to a fuzzy set in this paper we first remark that this measure of roughness has several undesirable properties and then propose a parameter free roughness measure for fuzzy sets based on the notion of the mass assignment of a fuzzy set several interesting properties of this new measure are examined furthermore we also discuss how the proposed approach is used to describe the rough approximation quality of a fuzzy classification

a general model of decision theoretic three way approximations of fuzzy sets based on a heuristic algorithm abstract a model of decision theoretic three way approximations of fuzzy sets that can calculate a pair of interpretable thresholds by combining shadowed sets with three way decisions was established by deng in that model the significant contribution is that the value 0 5 which denotes a membership grade with the highest uncertainty is used to replace the uncertain region with the interval 0 1 in shadowed sets from the principle of the minimum decision cost although the lowest overall cost can be achieved precisely in some data distributions based on the value 0 5 there is a difference between the obtained overall cost and the least cost in some cases therefore in this paper based on deng s model the concept of a general three way approximation of a fuzzy set is proposed to replace 0 5 with a variable value c 0