hidden sl n symmetry in conformal field theories we prove that an irreducible representation of the virasoro algebra can be extracted from an irreducible representation space of thesl 2 current algebra by putting a constraint on the latter using the becchi rouet stora tyutin formalism thus there is asl 2 symmetry in the virasoro algebra but it is gauged and hidden this construction of the virasoro algebra is the quantum analogue of the hamiltonian reduction we then are naturally lead to consider a constrainedsl 2 wess zumino witten model this system is also related to quantum field theory of coadjoint orbit of the virasoro group based on this result we present a canonical derivation of thesl 2 current algebra in polyakov s theory of two dimensional gravity it is a manifestation of thesl 2 symmetry in conformal field theory hidden by the quantum hamiltonian reduction we also discuss the quantum hamiltonian reduction of thesl 2 current algebra and its relation to thew n algebra of zamolodchikov this makes it possible to define a natural generalization of the geometric action for thew n algebra despite its non lie algebraic nature

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

a class of fuzzy clusterwise regression models in this paper we introduce a class of fuzzy clusterwise regression models with lr fuzzy response variable and numeric explanatory variables which embodies fuzzy clustering into a fuzzy regression framework the model bypasses the heterogeneity problem that could arise in fuzzy regression by subdividing the dataset into homogeneous clusters and performing separate fuzzy regression on each cluster the integration of the clustering model into the regression framework allows us to simultaneously estimate the regression parameters and the membership degree of each observation to each cluster by optimizing a single objective function the class of models proposed here includes as special cases the fuzzy clusterwise linear regression model and the fuzzy clusterwise polynomial regression model we also introduce a set of goodness of fit indices to evaluate the fit of the regression model within each cluster as well as in the whole dataset finally we consider some cluster validity criteria that are useful in identifying the optimal number of clusters several applications are provided in order to illustrate the approach

sugawara operators and kac kazhdan conjecture soit g une algebre de lie de kac moody affine de type x l 1 soit l le module de poids le plus eleve unique irreductible pour chaque poids de kac kazhdan on a chl exp re 1 exp 1

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

the revised method of ranking fuzzy numbers with an area between the centroid and original points in 2002 chu and tsao proposed a method to rank fuzzy numbers they employed an area between the centroid and original points to rank fuzzy numbers however there were some problems with the ranking method in this paper we want to indicate these problems of chu and tsao s method and then propose a revised method which can avoid these problems for ranking fuzzy numbers since the revised method is based on the chu and tsao s method it is easy to rank fuzzy numbers in a way similar to the original method

ranking fuzzy numbers with an area method using radius of gyration ranking fuzzy numbers plays an very important role in linguistic decision making and some other fuzzy application systems many methods have been proposed to deal with ranking fuzzy numbers chu pointed out some shortcomings of the existing distance method and proposed to rank the fuzzy numbers with the area between the centroid point and original point however drawbacks are also found in the area method for example it cannot rank fuzzy numbers when some fuzzy numbers have the same centroid point in this paper we propose a modified area method to rank fuzzy numbers the modified method can effectively rank various fuzzy numbers and their images we also used some comparative examples to illustrate the advantage of the proposed method

affine kac moody algebras and semi infinite flag manifolds we study representations of affine kac moody algebras from a geometric point of view it is shown that wakimoto modules introduced in 18 which are important in conformal field theory correspond to certain sheaves on a semi infinite flag manifold with support on its schhubert cells this manifold is equipped with a remarkable semi infinite structure which is discussed in particular the semi infinite homology of this manifold is computed the cousin grothendieck resolution of an invertible sheaf on a semi infinite flag manifold gives a two sided resolution of an irreducible representation of an affine algebras consisting of wakimoto modules this is just the brst complex as a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent lie algebra

applying the fuzzy weighted average approach to evaluate network security systems imprecise or vague importance as well as satisfaction levels of criteria may characterize the evaluation of alternative network security systems and may be treated by the fuzzy set theory as fuzzy numbers are adopted the fuzzy weighted averages fwas may become a suitable computation for the fuzzy criteria ratings and fuzzy weightings in this paper we propose to use the fwa approach that operates on the fuzzy numbers and obtains the fuzzy weighted averages during the evaluations of network security systems the algorithm constructs a benchmark adjustment solution approach for fwa a case study of the evaluation of network security systems is provided and demonstrates the usefulness of this study

area ranking of fuzzy numbers based on positive and negative ideal points the maximizing set and minimizing set method is a popular ranking approach for fuzzy numbers which ranks them based on their left right and total utilities this paper presents an alternative ranking approach for fuzzy numbers called area ranking based on positive and negative ideal points which defines two new alternative indices for the purpose of ranking the two new indices are defined in terms of a decision maker dm s attitude towards risks and the left and the right areas between fuzzy numbers and the two ideal points it is shown that the area ranking approach has strong discrimination power and can rank fuzzy numbers that are unable to be discriminated by the maximizing set and minimizing set method it is also shown that the dm s attitude towards risks may have a significant impact on the ranking of fuzzy numbers as a side product a new defuzzification formula is also developed and discussed

structure of some categories of representations of infinite dimensional lie algebras the highest weight representations of a finite dimensional complex semisimple lie algebra 9 have been studied extensively and a number of interesting results have been obtained in the last few years an extensive bibliography is given in 7 in 2 3 a category cy of representations of g was introduced which turns out to be the proper setting for considering various questions regarding highest weight representations the aim of this paper is to extend the study of this notion to a kac moody algebra or more generally to a contragredient lie algebra g a corresponding to a square matrix a over c the study of such algebras was begun independently in 8 14 and subsequently a number of interesting results were proved these results show that many important properties from the finite dimensional setup i e from the context of 9 can be proved in the general setup involving g a in particular the notion of category was extended in 9 to the case of kac moody algebras and used in the proof of combinatorial identities

fock representations of the affine lie algebra a 1 1 the aim of this note is to show that the affine lie algebraa1 1 has a natural family v of fock representations on the spacec xi yj i andj parametrized by v c2 by corresponding the highest weight of to each the parameter spacec2 forms a double cover of the weight spacec 0 c 1 with singularities at linear forms of level 2 this number is 1 times the dual coxeter number our results contain explicit realizations of irreducible non integrable highest wieghta1 1 modules for generic v

inconsistent property of lee and li fuzzy ranking method abstract this paper analyzes the lee and li fuzzy ranking method for the normalized trapezoidal fuzzy numbers ntfns every ntfn has associated with it another ntfn which posseses an inconsistent property an inconsistent property is that the results of comparison of these two ntfns are conflicting when two different comparison indices are used based on this property lee and li fuzzy ranking method suffers from lack of discrimination and occasionally conflicts with intuition

interval regression by tolerance analysis approach in interval linear regression analysis we are given crisp or interval data and we are to determine appropriate interval regression parameters there are various methods for interval regression many of them possess the property that while some of the resulting interval regression parameters are very wide the other parameters are crisp this drawback is the main limiting factor for such methods and much effort has been devoted to overcoming it we propose a method motivated by tolerance analysis in linear systems our method yields intervals for regression parameters the widths of which are proportional to an in advance given vector of parameters moreover the method is computationally very cheap and provides a natural measure of quality of a model first we formulate the method for the basic model of crisp input crisp output data and then extend it to crisp input interval output and interval input interval output models for the interval valued cases we study several formulations of the solution concept possibility strong possibility weak possibility necessity here strong possibility is a new concept proposed as a natural counterpart to the remaining ones we prove that the method provides optimal interval parameters meeting centrality and proportionality requirements we also show that the method provides interval regression parameters satisfying various versions of tanaka lee s inclusion property we also derive a form of a complementarity theorem for the weak possibility and necessity solution concepts since practical problems may be affected by outlying observations we show that our approach is easily adapted to deal with them we illustrate the theory by examples

routing and wavelength assignment by partition colouring the problem of routing and wavelength assignment in all optical networks may be solved by a combined approach involving the computation of alternative routes for the lightpaths followed by the solution of a partition colouring problem in a conflict graph a new tabu search heuristic is also proposed for the partition colouring problem which may be viewed as an extension of the graph colouring problem computational experiments are reported showing that the tabu search heuristic outperforms the best heuristic for partition colouring by approximately 20 in the average and illustrating that the new approach for the problem of routing and wavelength assignment is more robust than a well established heuristic for this problem

ranking alternatives using fuzzy numbers we investigate the problem of employing expert opinion to rank alternatives across a set of criteria the experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue s fuzzy ranking this leads to a partition of the alternatives into sets h1 h2 where h1 contains the highest ranked issues h2 has all the second highest ranked alternatives etc the total ranking process is shown to possess a number of important properties an example is presented to illustrate the method

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

ranking multi criterion river basin planning alternatives using fuzzy numbers abstract the methodology proposed by anand raj and nagesh kumar 5 to rank the river basin planning and development alternatives under multi criterion environment using fuzzy numbers is applied to a case study the purpose is to find the most suitable planning of reservoirs with their associated purposes aimed at the development of one of the major peninsular river basins krishna river basin in india a set of 7 alternative systems with 8 main objectives which are further subdivided into 18 criteria are considered for ordering or ranking them employing the opinion preference structure of three experts an acadamician a field engineer and an official from ministry of water resources using fuzzy numbers the fuzzy weights w i of alternatives a i are computed using standard fuzzy arithmetic the concepts of maximizing set and minimizing set are introduced to decide total utility or order value of each of the alternatives

ranking nonnormal p norm trapezoidal fuzzy numbers with integral value this paper considers the ranking fuzzy numbers with integral value proposed by liou and wang for the nonnormal p norm trapezoidal fuzzy numbers two interesting special cases of p norm trapezoidal fuzzy numbers are the well known triangular and trapezoidal fuzzy numbers for the nonnormal fuzzy numbers the differences exist among the membership functions for the triangular trapezoidal and p norm trapezoidal fuzzy numbers these differences can affect their left right and total integral values so we establish the relationship between these three types of fuzzy numbers

a fuzzy multi criteria decision making model by associating technique for order preference by similarity to ideal solution with relative preference relation generally classical multi criteria decision making mcdm methods were extended to encompass uncertainty and vagueness of messages under fuzzy environment for solving decision making problems especially for technique for order preference by similarity to ideal solution topsis in the fuzzy extension of topsis fuzzy numbers comparison and aggregation based on fuzzy preference relation are important issues to compute distance values between alternatives and ideal or anti ideal solution or rank feasible alternatives because lots of messages are reserved by fuzzy preference relation however fuzzy preference relation on pair wise comparison is commonly too complex to calculate to avoid the drawback we use a relative preference relation improved from fuzzy preference relation in the fuzzy extension of topsis for computing distance values between alternatives and ideal or anti ideal solution or obtaining relative closeness coefficients of alternatives thus the relative preference relation on fuzzy numbers will be associated with topsis under fuzzy environment to develop a fuzzy multi criteria decision making fmcdm model through the association above fmcdm problems can be easily solved by the model further we compare the proposed model with other methods to demonstrate the model s feasibility and rationality

a subjective approach for ranking fuzzy numbers abstract starting from a subjective assignation of weights related to the relative importance of the different level sets we define a new method of comparison of fuzzy numbers this procedure is an extension of some well known indices adamo tsumura et al yager some properties of our index are studied in this paper as well as its behaviour on several particular cases

a heuristic approach for combined equipment planning and routing in multi layer sdh wdm networks abstract the paper deals with a multi layer network design problem for a high speed telecommunication network based on synchronous digital hierarchy sdh and wavelength division multiplex wdm technology the network has to carry a certain set of demands with the objective of minimizing the investment in the equipment the different layers are the fiber layer 2 5 gbit s 10 gbit s and wdm systems several variations of the problem including path protected demands and specific types of cross connect equipment are considered the problem is described as a mixed integer linear programming model and some results for small networks are presented two greedy heuristics a random start heuristic and a grasp like approach are implemented to solve large real world problems

ranking alternatives with fuzzy weights using maximizing set and minimizing set ranking alternatives both qualitative as well as quantitative in a multicriterion environment employing experts opinion preference structure using fuzzy numbers and linguistic variables are presented in this paper fuzzy weights wj i of alternatives a i are computed using standard fuzzy arithmetic concept of maximizing set and minimizing set is introduced to decide total utility or ordering value of each of the alternatives a numerical example is provided at the end to illustrate the method 1999 published by elsevier science b v all rights reserved

generalizing topsis for fuzzy multiple criteria group decision making in this paper we generalize topsis to fuzzy multiple criteria group decision making fmcgdm in a fuzzy environment topsis is one of the well known methods for multiple criteria decision making mcdm most of the steps of topsis can be easily generalized to a fuzzy environment except max and min operations in finding the ideal solution and negative ideal solution thus we propose two operators up and lo which satisfy the partial ordering relation on fuzzy numbers to the generalization of topsis in generalized topsis these two operations up and lo are employed to find ideal and negative ideal solutions under a fuzzy environment then the fmcgdm problem can be solved effectively and efficiently

a class of non standard modules for affine lie algebras on etudie la structure des modules de verma pour des algebres de lie affines quand le poids le plus haut est orthogonal aux racines imaginaires on considere l algebre de lie a 1 1 sl 2 c 1

the multiple judge multiple criteria ranking problem a fuzzy set approach abstract this paper investigates the problem of selecting from a set of issues those which best satisfy a collection of criteria a group of judges have fuzzy sets defined over the issues for each criterion whose values lie in a finite linearly ordered set l these judges also have fuzzy sets defined over the set of criteria the paper discusses methods of aggregating all the fuzzy sets into one fuzzy set defined on the issues so that a l gives the final ranking for issue a

interval valued fuzzy multi criteria decision making based on simple additive weighting and relative preference relation abstract to encompass imprecise messages traditional multi criteria decision making mcdm was generalized into fuzzy multi criteria decision making fmcdm in solving decision making problems in numerous mcdm methods simple additive weighting saw is the simplest one but fuzzy multiplication for generalizing saw under fuzzy environment is still complicated especially interval valued fuzzy numbers in the past wang proposed a model based on saw and relative preference relation to resolve fuzzy multiplication tie practically wang s model is useful to fuzzy generalization of saw however the model is only used in triangular fuzzy numbers recently more and varied messages are obtained and processed for decision making problems triangular fuzzy numbers are insufficient to express the variance whereas interval valued fuzzy numbers are suitable therefore saw had to be generalized under interval valued fuzzy environment in this paper we combine saw with a relative preference relation into interval valued fmcdm to avoid fuzzy multiplication and grasp more messages our proposed relative preference relation improved from lee s fuzzy preference relation is similar to wang s relative preference relation the main difference is that the proposed relation is utilized in interval valued fuzzy numbers whereas wang s is used in triangular fuzzy numbers to sum up we provide the fmcdm to solve decision making problems with interval valued fuzzy numbers easily and quickly

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

ranking fuzzy numbers with an area between the centroid point and original point abstract to improve the ranking method of lee and li 1 cheng 2 proposed the coefficient of variation cv index shortcomings are also found in the cv index cheng 2 also proposed the distance method to improve the ranking method of murakami et al however the distance method is not sound either moreover the cv index contradicts the distance method in ranking some fuzzy numbers therefore to overcome the above shortcomings we propose ranking fuzzy numbers with the area between the centroid point and original point

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

ranking fuzzy numbers with maximizing set and minimizing set abstract up to now these are five methods of ranking n fuzzy numbers in order but these methods contain some confusions and occasionally conflict with intuition this paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers in addition we give a method for calculating the ordering value of each fuzzy number with triangular trapezoidal and two sided drum like shaped membership functions

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

a dual ascent approach for steiner tree problems on a directed graph the steiner tree problem on a directed graph stdg is to find a directed subtree that connects a root node to every node in a designated node setv we give a dual ascent procedure for obtaining lower bounds to the optimal solution value the ascent information is also used in a heuristic procedure for obtaining feasible solutions to the stdg computational results indicate that the two procedures are very effective in solving a class of stdg s containing up to 60 nodes and 240 directed 120 undirected arcs