axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

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

weighted scale free networks with variable power law exponents abstract we present a weighted scale free network model in which the power law exponents can be controlled by the model parameters the network is generated through the weight driven preferential attachment of new nodes to existing nodes and the growth of the weights of existing links the simplicity of the model enables us to derive analytically the various statistical properties such as the distributions of degree strength and weight the degree strength and degree weight relationship and the dependencies of these power law exponents on the model parameters finally we demonstrate that networks of words coauthorship of researchers and collaboration of actor actresses are quantitatively well described by this model

a comprehensive study of implicator conjunctor based and noise tolerant fuzzy rough sets definitions properties and robustness analysis abstract both rough and fuzzy set theories offer interesting tools for dealing with imperfect data while the former allows us to work with uncertain and incomplete information the latter provides a formal setting for vague concepts the two theories are highly compatible and since the late 1980s many researchers have studied their hybridization in this paper we critically evaluate most relevant fuzzy rough set models proposed in the literature to this end we establish a formally correct and unified mathematical framework for them both implicator conjunctor based definitions and noise tolerant models are studied we evaluate these models on two different fronts firstly we discuss which properties of the original rough set model can be maintained and secondly we examine how robust they are against both class and attribute noise by highlighting the benefits and drawbacks of the different fuzzy rough set models this study appears a necessary first step to propose and develop new models in future research

branching annihilating random walk on random regular graphs the branching annihilating random walk is studied on a random graph whose sites have a uniform number of neighbors z the monte carlo simulations in agreement with the generalized mean field analysis indicate that the concentration decreases linearly with the branching rate for z 4 while the coefficient of the linear term becomes zero if z 3 these properties are described by a modified mean field theory taking explicitly into consideration the probability of mutual annihilation of the parent and its offspring particles using the returning features of a single walker on the same graph

holder estimates on cr manifolds with a diagonalizable levi form a for all e 0 the operator cl maps lip s po into lip s 2 m e po here lip s p denotes the space of square integrable forms of degree p q with 1 g q otheoperators o l o l q and0 map lip s p into lip s l m e po c for all e o the operators 8b8zo 8z8ho 1 q a aa cl 8 8 8 q 8 and 8 0 ab map lip s p into lip s p hebre aiag web asm acts on forms of type p q with 1 q 0 e zf f is a square integrable function on a with f e lip p then s f e lip s e p for all e 0 where s is the orthogonal projection on square integrable cr functions

a parametrix for the neumann problem on pseudoconvex domains of finite type abstract we construct a parametrix for the neumann problem on any pseudoconvex domain of finite type which reduces completely the study of isotropic lp sobolev and holder estimates to those for b and b we also establish the maximal nonisotropic estimates in the case of d q domains and provide a precise comparison of the bergman and szego projections and of the canonical solutions to with respect to these two operators the argument is substantially simpler than the known ones for the cases of strongly pseudoconvex domains and pseudoconvex domains of finite type in c 2

hearing the type of a domain in c 2 with the neumann laplacian a smooth bounded pseudoconvex domain in c2 is of finite type if and only if the number of eigenvalues of its a neumann laplacian that are less than or equal to has at most polynomial growth as goes to infinity

what is the real size of a sampled network the case of the internet internet mapping projects generally consist in sampling the network from a limited set of sources by usingtraceroute probes this methodology akin to the merging of spanning trees from the different sources to a set of destinations leads necessarily to a partial incomplete map of the internet accordingly determination of internet topology characteristics from such sampled maps is in part a problem of statistical inference our contribution begins with the observation that the inference of many of the most basic topological quantities including network size and degree characteristics fromtraceroute measurements is in fact a version of the so called species problem in statistics this obse rvation has important implications since species problems are often quite challenging we focus here on the most fundamental example of atraceroute internet species the number of nodes in a network specifica lly

the szegő projection sobolev estimates in regular domains the szego projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of diederich fornxess and catlin it preserves local smoothness near boundary points of finite type the bergman projection being linked to the 8 problem plays an important role in several complex variables as demonstrated by recent applications to the boundary behavior of holomorphic maps 7 16 the szego projection is interesting both by analogy and by its connection with the ab problem and hence with unsolvable equations of lewy type while in strictly pseudoconvex domains the bergman and szego projections are both well understood in general much more is known about the bergman projection this is no surprise for the theory of the 8 problem is further advanced than the theory of the 8b problem nonetheless i believe that the szego projection ought not be any more mysterious than the bergman projection this paper is intended to support this contention the most important domains for which the bergman projection is known to be regular are domains with a good a neumann operator here it is shown that in such domains the szego projection also is regular more precisely in smooth bounded pseudoconvex domains satisfying catlin s property p the most general condition presently known that guarantees global regularity of the neumann operator the szego projection is shown to be globally regular theorem 4 1 moreover at boundary points of finite type the szego projection is locally regular theorem 5 1 in both cases the szego projection exactly preserves the degree of differentiability measured in sobolev norms the method of proof is not the obvious one of directly generalizing from the strictly peudoconvex case for there are serious obstacles to such a generalization for instance the approach of analysis on the heisenberg group used by phong and stein 29 to derive estimates for both the bergman and szego projections cannot yet be extended because the theory of singular integrals on weakly pseudoconvex surfaces is still in its early stages 28 15 similarly one does not know how to carry over the fourier integral operator approach of boutet de monvel and sjostrand 10 to the weakly pseudoconvex case explicit construction of cauchy fantappie forms received by the editors march 11 1986 paper presented april 11 1986 to the 826th meeting of the american mathematical society indianapolis special session on several complex variables 1980 mathematics subject classification 1985 revision primary 32a25 32h10 partially supported by national science foundation grants mcs 8201063 and dms 8501758 1987 american mathematical society 0002 9947 87 1 00 25 per page

epidemic incidence in correlated complex networks a few years ago watts and strogatz 1 introduced a model able to produce networks with properties of both regu lar lattices and random graphs with small diameter their model soon led to a burst of activity in the field 2 3 further spurred by barabasi and co workers who found that many seemingly diverse systems share several topological proper ties such as a power law behavior in their connectivity dis tributions when represented as networks 4 these complex networks are formed by a set of many elements or nodes that are linked together through edges or links if they in teract directly empirical evidence supports that in notable networks such as metabolic or communication webs the probability p k that any node has k links to other nodes is distributed accordingly to a power law p k k 2g 5 7

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

necessary geometric and analytic conditions for general estimates in the bar partial neumann problem we show the geometric and analytic consequences of a general estimate in the bar partial neumann problem a gain in the estimate yields a bound in the type of the boundary that is in its order of contact with an analytic curve as well as in the rate of the bergman metric we also discuss the potential theoretical consequence a gain implies a lower bound for the levi form of a bounded weight

construction of p s h functions on weakly pseudoconvex domains let q c c be a smoothly bounded pseudoconvex domain and let a q denote the functions holomorphic on q and continuous on sl a point p e aq is a peak point if there is a function f e a q such that f p 1 and if z i 2 were obtained by range 19 20 and hakim and sibony 12 these results do not extend automatically to arbitrary weakly pseudoconvex boundary points of q even if the boundary is real analytic an example of kohn and nirenberg 17 shows that a p e aq does not always have a separating function which is real analytic at p similarly this example does not have a peak function which is e at p see fornaess 6 here we show how peak functions and separating functions may be constructed for a large class of weakly pseudoconvex points on domains in c the results are most complete for domains in c2 so we state them here for that case

hearing pseudoconvexity with the kohn laplacian a bounded domain in open image in new window with connected lipschitz boundary is pseudoconvex if the bottom of the essential spectrum of the kohn laplacian on the space of 0 q forms 1 q n 1 with l2 coefficients is positive

defining equations for real analytic real hypersurfaces in ⁿ a defining function for a real analytic real hypersurface can be utliquely written as 2 re h e where h is holomorphic and e contains no pure terms we study how h and e change when we perform a local biholomorphic change of coordinates or multiply by a unit one of the main results is necesary and sufficient conditions on the first nonvanishing homogeneous part of e expanded in terms of h beyond eoo that serve as obstructions to writing a defining equation as 2 re h e where e is independent of h we also find necessary pluriharmonic obstructions to doing this which arise from the easier case of attempting to straighten the hypersurface

rough sets and fuzzy sets some remarks on interrelations abstract this short paper refers to two remarks given in 3 and disputes them any rough set can be expressed by means of a membership function u 0 0 5 1 we show that this representation can be extended to union and intersection

scale free networks via attaching to random neighbors abstract preferential attachment is considered one of the key factors in the formation of scale free networks however complete random attachment without a preferential mechanism can also generate scale free networks in nature such as protein interaction networks in cells this article presents a new scale free network model that applies the following general mechanisms i networks expand continuously by the addition of new vertices and ii new vertices attach to random neighbors of random vertices that are already well connected the proposed model does not require global based preferential strategies and utilizes only the random attachment method theoretical analysis and numerical simulation results denote that the proposed model has steady scale free network characteristics and random attachment without a preferential mechanism may generate scale free networks

compactness in the neumann problem magnetic schrodinger operators and the aharonov bohm effect abstract compactness of the neumann operator in the neumann problem is studied for weakly pseudoconvex bounded hartogs domains in two dimensions a nonsmooth domain is constructed for which condition p fails to hold yet the kohn laplacian still has compact resolvent the main result in contrast is that for smoothly bounded hartogs domains the well known sufficient condition p is equivalent to compactness the analyses of compactness and condition p boil down to the asymptotic behavior of the lowest eigenvalues of two related sequences of schrodinger operators one with a magnetic field and one without parametrized by a fourier variable resulting from the hartogs symmetry the nonsmooth example is based on the aharonov bohm phenomenon of quantum physics for smooth domains not satisfying p we prove that there always exists an exceptional sequence of fourier variables for which the aharonov bohm effect is weak and thence that compactness fails to hold this sequence can be very sparse so that the lack of compactness is due to a rather subtle effect

epidemic dynamics behavior in some bus transport networks abstract we abstract bus transport networks btns to complex networks using the space p approach first we select three actual btns in three major cities in china namely beijing shanghai and hangzhou using the sis model we simulate and study the epidemic spreading in the three btns we obtain the density of infected vertices varying with time and the stationary density of infected vertices varying with infection rate second we simulate and study the epidemic spreading in a recently introduced btn evolution model the network properties of which correspond well with those of actual btns third we use mean field theory to analyze the epidemic dynamics behavior of the btn evolution model and obtain the theoretical epidemic threshold of this model the theoretical value agrees well with the simulation results based on the work in this paper we provide the following possible forecasts for epidemic dynamics in actual btns an actual btn should have a finite positive epidemic threshold if the effective infection rate is above this threshold the epidemic spread in the network and the density of infected vertices finally stabilizes in a balanced state below this threshold the number of infected vertices decays exponentially fast and the epidemic cannot spread on a large scale

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

the bergman kernel on certain weakly pseudoconvex domains let c c be a bounded pseudoconvex domain with c p lur i superharmonic defining function 0 let az q be the space of square integrable ho lomorph ic functions on if 0 is any complete o r thonorma l basis for az g2 then the bergman kernel for o is given by k z w y o z o w except for some special cases 53 it is not possible at present to obta in a nice closed form for k however for domains of the form

compactness of the complex green operator on cr manifolds of hypersurface type the purpose of this article is to study compactness of the complex green operator on cr manifolds of hypersurface type we introduce cr pq a potential theoretic condition on 0 q forms that generalizes catlin s property pq to cr manifolds of arbitrary codimension we prove that if an embedded cr manifold of hypersurface type of real dimension at least five satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 and s 0 we use cr plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle 1 introduction and results in this article we introduce property cr pq a potential theoretic condition on 0 q forms we show that if an embedded cr manifold of hypersurface type satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 we use cr plurisubharmonic functions to build a microlocal norm that controls the bad direction of the tangent bundle we first prove the closed range and compactness results on l 2 q m and use an elliptic regularization

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

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

on characterizations of i t fuzzy rough approximation operators in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties various fuzzy generalizations of rough approximations have been made in the literature this paper proposes a general framework for the study of i t fuzzy rough approximation operators within which both constructive and axiomatic approaches are used in the constructive approach a pair of lower and upper generalized fuzzy rough approximation operators determined by an implicator i and a triangular norm t is first defined basic properties of i t fuzzy rough approximation operators are investigated the connections between fuzzy relations and fuzzy rough approximation operators are further established in the axiomatic approach an operator oriented characterization of rough sets is proposed that is i t fuzzy approximation operators are defined by axioms different axiom sets of t upper and i lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators finally an open problem proposed by radzikowska and kerre in fuzzy sets and systems 126 2002 137 is solved

necessary conditions for subellipticity of the overline partial neumann problem let a be a a closed form of type p q with l2 coefficients on a smoothly bounded domain q in c the problem of determining both the existence and regularity properties of the solution u of au a where u is orthogonal to the null space of a on p q l forms is known as the a neumann problem one of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates let u be a neighborhood of a point z0 in the boundary of q a subelliptic estimate is said to hold in u if the estimate

convex domains of finite type we construct bounded plurisubharmonic functions with maximally large hessians near the boundary of a smoothly bounded convex domain in cn as a corollary the equality of the order of contact of the boundary with complex analytic varieties the d angelo type and the order of contact with complex lines is demonstrated

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

research on the search ability of brownian particles on networks with an adaptive mechanism abstract in this paper we focus on the search ability of brownian particles with an adaptive mechanism in the adaptive mechanism nodes are allowed to be able to change their own accepting probability according to their congestion states two searching traffic models the static one in which nodes have fixed accepting probability to the incoming particles and the adaptive one in which nodes have adaptive accepting probability to the incoming particles are presented for testing the adaptive mechanism instead of number of hops we use the traveling time which includes not only the number of hops for a particle to jump from the source node to the destination but also the time that the particle stays in the queues of nodes to evaluate the search ability of brownian particles we apply two models to different networks the experiment results show that the adaptive mechanism can decrease the network congestion and the traveling time of the first arriving particle furthermore we investigate the influence of network topologies on the congestion of networks by addressing several main properties degree distribution average path length and clustering coefficient we show the reason why random topologies are more able to deal with congested traffic states than others we also propose an absorption strategy to deal with the additional brownian particles in networks the experiment results on barabasi albert ba scale free networks show that the absorption strategy can increase the probability of a successful search and decrease the average per node particles overhead for our models

effective vanishing order of the levi determinant on a smooth domain in mathbb c n of finite d angelo q type at a point an effective upper bound for the vanishing order of the levi determinant rm coeff partial r wedge bar partial r wedge partial bar partial r n q at that point is given in terms of the d angelo q type the dimension of the space n and q itself the argument uses catlin s notion of a boundary system as well as techniques pioneered by john d angelo

statistical analysis of 22 public transport networks in poland public transport systems in 22 polish cities have been analyzed the sizes of these networks range from n 152 to 2881 depending on the assumed definition of network topology the degree distribution can follow a power law or can be described by an exponential function distributions of path lengths in all considered networks are given by asymmetric unimodal functions clustering assortativity and betweenness are studied all considered networks exhibit small world behavior and are hierarchically organized a transition between dissortative small networks n approximately or 500 is observed

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

twofold fuzzy sets and rough sets some issues in knowledge representation abstract this paper deals with the representation of sets where the membership of some elements may be ill known rather than just a matter of degree as in a fuzzy set the notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete a twofold fuzzy set is made of a nested pair of fuzzy sets the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by pawlak set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined twofold fuzzy relations are also introduced finally various applications of twofold fuzzy sets in knowledge representation are briefly discussed

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

a study on some urban bus transport networks abstract in this paper we present the empirical investigation results on the urban bus transport networks btns of four major cities in china in btn nodes are bus stops two nodes are connected by an edge when the stops are serviced by a common bus route the empirical results show that the degree distributions of btns take exponential function forms other two statistical properties of btns are also considered and they are suggested as the distributions of so called the number of stops in a bus route represented by s and the number of bus routes a stop joins by r the distributions of r also show exponential function forms while the distributions of s follow asymmetric unimodal functions to explain these empirical results and attempt to simulate a possible evolution process of btn we introduce a model by which the analytic and numerical result obtained agrees well with the empirical facts finally we also discuss some other possible evolution cases where the degree distribution shows a power law or an interpolation between the power law and the exponential decay

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

scaling and correlations in three bus transport networks of china abstract we report the statistical properties of three bus transport networks btn in three different cities of china these networks are composed of a set of bus lines and stations serviced by these network properties including the degree distribution clustering and average path length are studied in different definitions of network topology we explore scaling laws and correlations that may govern intrinsic features of such networks besides we create a weighted network representation for btn with lines mapped to nodes and number of common stations to weights between lines in such a representation the distributions of degree strength and weight are investigated a linear behavior between strength and degree s k k is also observed