a fuzzy compromise approach to water resource systems planning under uncertainty abstract a fuzzy compromise approach to decision analysis is described within the context of water resource systems planning under uncertainty the approach allows various sources of uncertainty and is intended to provide a flexible form of group decision support the example compares the electre method with the fuzzy compromise approach the comparison is intended to demonstrate the benefits of adopting a multicriteria decision analysis technique which presents subjectivity within its proper context while maintaining an intuitive and transparent technique for ranking alternatives the fuzzy compromise approach allows a family of possible conditions to be reviewed and supports group decisions through fuzzy sets designed to reflect collective opinions and conflicting judgements ranking of alternatives is accomplished with fuzzy ranking measures designed to illustrate the effect of risk tolerance differences among decision makers two distinct ranking measures are used a centroid measure and a fuzzy comparison measure based on a fuzzy goal

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

unitary representations and theta correspondence for type i classical groups in this paper we discuss the positivity of the hermitian form p introduced by li in invent math 27 1989 237 255 let g1 g2 be a type i dual pair with g1 the smaller group let p be an irreducible unitary representation in the semistable range of y mg1 mg2 see communications in contemporary mathematics vol 2 2000 pp 255 283 we prove that the invariant hermitian form p is positive semidefinite under certain restrictions on the size of g2 and a mild growth condition on the matrix coefficients of p therefore if p does not vanish y mg1 mg2 p is unitary theta correspondence over r was established by howe in j amer math soc 2 1989 535 552 li showed that theta correspondence preserves unitarity for dual pairs in stable range our results generalize the results of li for type i classical groups invent math 27 1989 237 the main result in this paper can be used to construct irreducible unitary

reasonable properties for the ordering of fuzzy quantities ii as a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches to do so we briefly introduce fuzzy relations on which the third class of approaches is based then we recall some transitivity related concepts and an ordering procedure based on a acyclic fuzzy relation acyclicity is a very weak restriction on a fuzzy relation we prove that many fuzzy relations used for the comparison of fuzzy quantities satisfy some conditions stronger than acyclicity so we give a widely applicable formulation to derive a total ranking order from a fuzzy relation with our formulation we examine all the ordering indices in the third class with respect to the proposed axioms in part i

ranking fuzzy numbers with integral value abstract ranking fuzzy numbers is important in decision making since very often the alternatives are evaluated by fuzzy numbers in a vague environment a comparison between these fuzzy numbers is indeed a comparison between alternatives this paper proposes a method of ranking fuzzy numbers with integral value the method which is independent of the type of membership functions used and the normality of the functions can rank more than two fuzzy numbers simultaneously it is relatively simple in computation especially in ranking triangular and trapezoidal fuzzy numbers further an index of optimism is used to reflect the decision maker s optimistic attitude discussion on comparative advantages is included

a probability based approach to comparison of fuzzy numbers and applications to target oriented decision making in this paper we introduce a new comparison relation on fuzzy numbers based on their alpha cut representation and comparison probabilities of interval values basically this comparison process combines a widely accepted interpretation of fuzzy sets together with the uncertain characteristics inherent in the representation of fuzzy numbers the proposed comparison relation is then applied to the issue of ranking fuzzy numbers using fuzzy targets in terms of target based evaluations some numerical examples are used to illuminate the proposed ranking technique as well as to compare with previous methods more interestingly according to the interpretation of the new comparison relation on fuzzy numbers we provide a fuzzy target based decision model as a solution to the problem of decision making under uncertainty with which an interesting link between the decision maker s different attitudes about target and different risk attitudes in terms of utility functions can be established moreover an application of the proposed comparison relation to the fuzzy target based decision model for the problem of fuzzy decision making with uncertainty is provided numerical examples are also given for illustration

the optimum output quantity of a duopoly market under a fuzzy decision environment the main purpose of this paper is to develop a new optimum output quantity decision analysis of a duopoly market under a fuzzy decision environment to efficiently handle the fuzziness of the decision variables the linguistic values subjectively represented by the trapezoidal fuzzy numbers are used to act as the evaluation tool of decision variables such as fixed cost and unit variable cost this paper will apply fuzzy set theory to construct an optimum output quantity decision model based on aiming for the maximum profit of a duopoly market by using this decision model the decision makers fuzzy assessments with various variables can be considered in the decision process to assure more convincing and accurate decision making

a method for ranking fuzzy numbers and its application to decision making since fuzzy numbers represent uncertain numeric values it is difficult to rank them according to their magnitude in the paper a method for ranking fuzzy numbers is proposed the method considers the overall possibility distributions of fuzzy numbers in their evaluations for ranking and provides users with a method of changing viewpoints for evaluations users represent their viewpoints with fuzzy sets the method evaluates fuzzy numbers with a satisfaction function and the viewpoint given by users and then ranks the numbers according to their evaluation values the satisfaction function is a measure of comparisons between fuzzy numbers in order to illustrate the ranking method two numeric examples are shown and for the comparative study our method is compared with four existing ranking methods through eight examples as an example of potential applications the proposed method is applied to a decision making problem a two person game with fuzzy profit and loss the ranking method is used to analyze player choices

a new method to rank fuzzy numbers using dempster shafer theory with fuzzy targets fuzzy numbers are ranked using dempster shafer theory dst with fuzzy targets fuzzy targets are used to reflect human viewpoints and avoid biases in fuzzy ranking fuzzy numbers are ranked with different viewpoints and with a combined viewpoint transferable belief model is adopted to elaborate dst benchmark examples are adopted for experimental studies and positive results are obtained in this paper an extended ranking method for fuzzy numbers which is a synthesis of fuzzy targets and the dempster shafer theory dst of evidence is devised the use of fuzzy targets to reflect human viewpoints in fuzzy ranking is not new however different fuzzy targets can lead to contradictory fuzzy ranking results making it difficult to reach a final decision in this paper the results from different viewpoints are treated as different sources of evidence and murphy s combination rule is used to aggregate the fuzzy ranking results dst allows fuzzy numbers to be compared and ranked while preserving their uncertain and imprecise characteristics in addition a hybrid method consisting of fuzzy targets and dst with the transferable belief model is formulated which fulfils a number of important ordering properties a series of empirical experiments with benchmark examples has been conducted and the experimental results clearly indicate the usefulness of the proposed method

construction of global fuzzy preference structures from two parameter single criterion fuzzy outranking relations in this paper we deal with the construction of global fuzzy preference structures from two parameter single criterion fuzzy outranking relations proposed in 10 and which are based on the aggregation of outranking indices associated with a cuts we consider two ways of constructing global fuzzy preference structures i aggregate the two parameter single criterion fuzzy outranking relations and then construct the global fuzzy preference structures and ii construct global fuzzy preference structures and then aggregate the single criterion fuzzy preference indifference and incomparability relations we also investigate the case where these two ways of proceeding lead to the same results

ranking of fuzzy sets based on the concept of existence abstract various approaches have been proposed for the comparison or ranking of fuzzy sets however due to the complexity of the problem a general method which can be used for any situation still does not exist this paper formalizes the concept of existence for the ranking of fuzzy sets many of the existing fuzzy ranking methods are shown to be some application of this concept an improved fuzzy ranking method is then introduced based on this concept this newly introduced method is expanded for treating both normal and nonnormal convex and nonconvex fuzzy sets emphasis is placed on the use of the subjectivity of the decision maker such as the optimistic or the pessimistic view points an improved procedure for obtaining linguistic conclusions is also developed finally some numerical examples are given to illustrate the approach

on semisimple hopf algebras of dimension pq2 let p and q be distinct prime numbers we prove a result on the existence of nontrivial group like elements in a certain class of semisimple hopf algebras of dimension pqr we conclude the classification of semisimple hopf algebras a of dimension pq2 over an algebraically closed field k of characteristic zero such that both a and a are of frobenius type we also complete the classification of semisimple hopf algebras of dimension pq2 100

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

simplifying the search for effective ranking of fuzzy numbers we prove that a ranking index which generates an order on some subset of fuzzy numbers can be reduced to a ranking index with a simpler form and which generates an equivalent order this way the finding of ranking indices generating orders with desirable properties can be essentially simplified such that we need to search only from additive or scale invariant ranking indices we determine exactly making an abstraction of equivalent orders the class of ranking indices on trapezoidal fuzzy numbers which generate orders satisfying all the basic requirements finally ranking indices used to rank trapezoidal fuzzy numbers are extended to ranking indices used to rank arbitrary fuzzy numbers so that the desired properties are preserved

on howe s duality theorem abstract we adopt the langlands classification to the context of real reductive dual pairs and prove that howe s duality correspondence maps hermitian representations to hermitian representations

representations of quantum permutation algebras we develop a combinatorial approach to the quantum permutation algebras as hopf images of representations of type pi a s n to b h we discuss several general problems including the commutativity and cocommutativity ones the existence of tensor product or free wreath product decompositions and the tannakian aspects of the construction the main motivation comes from the quantum invariants of the complex hadamard matrices we show here that under suitable regularity assumptions the computations can be performed up to n 6

singular unitary representations of classical groups on presente une description de la partie relativement singuliere du dual unitaire d un groupe classique de type i sur un corps local f de caracteristique autre que 2

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

construction of a two parameters fuzzy outranking relation from fuzzy evaluations at the early stages of design process several design concepts are proposed as potential solutions the selection of the most appropriate one is based on their evaluation with respect to several conflicting criteria in this paper we consider the case where the evaluations of design concepts with respect to different criteria are represented by means of fuzzy numbers we deal with the construction of a two parameters fuzzy outranking relation from fuzzy evaluations it is based on the aggregation of outranking indices associated with alpha cuts this fuzzy outranking index allows the decision maker to adopt different strategies by choosing a degree of optimism and a degree of aggressiveness

ranking fuzzy numbers with preference weighting function expectations based on the recent research developments of fuzzy number ranking the paper extends the centroid expectation approach and proposes a preference weighting function expectation method to rank fuzzy numbers the weighting function gives different emphasis on the possible values of the fuzzy number which can be used to represent the decision maker s pessimistic neutral or optimistic attitude an optimistic measure to represent this preference information is proposed some properties of the weighting function and optimistic measure are systematically analyzed then the paper proposes a special class weighting function with maximum entropy principle and analyzes some of its properties with this parameterized maximum entropy weighting function we can get preference expectation of any fuzzy number with given optimistic level next the weighting function is extended to represent the confidential attitude of the decision maker by giving different emphasis on the cut level set of the fuzzy number some results about confidential attitude that parallel to the valuation attitude are attained we especially extended the badd basic defuzzification distribution approach with preference expectation of fuzzy numbers and another confidential measure with badd operator is proposed finally the expected valuations of fuzzy numbers are used to ranking the fuzzy numbers compared with other fuzzy number ranking methods the method of this paper is easily interpreted robust flexible and invariant with translating and scaling for real numbers

fuzzy decision making based on likelihood based comparison relations in this paper we present a new fuzzy decision making method which is based on likelihood based comparison relations first we introduce the concepts of likelihood based comparison relations for intervals then we propose the concept of likelihood based comparison relations for type 1 fuzzy sets and interval type 2 fuzzy sets then we present a new method to rank fuzzy sets by using fuzzy targets based on the proposed likelihood based comparison relations for fuzzy sets finally we present a new fuzzy decision making method based on the proposed likelihood based comparison relations for fuzzy sets and the proposed fuzzy ranking method the proposed fuzzy decision making method has the advantage that the evaluated values can either be represented by crisp values intervals type 1 fuzzy sets or interval type 2 fuzzy sets it can overcome the drawbacks of huynh et al s method due to the fact that huynh et al s method cannot deal with the ranking of interval type 2 fuzzy sets for fuzzy decision making and cannot distinguish the ranking order between the alternatives in some situations

compositions of theta correspondences theta correspondence y over r is established by howe j amer math soc 2 1989 535 in he j funct anal 199 2003 92 we prove that y preserves unitarity under certain restrictions generalizing the result of li invent math 97 1989 237 the goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences we show that under a natural condition on the sizes of the related dual pairs which can be predicted by the orbit method j algebra 190 1997 518 representation theory of lie groups park city 1998 pp 179 238 the orbit correspondence for real and complex reductive dual pairs preprint 2001 one can compose theta correspondences to obtain unitary representations we call this process quantum induction r 2004 published by elsevier inc msc primary 22e45 22e46

classification of semisimple hopf algebras of dimension 16 abstract in this paper we completely classify nontrivial semisimple hopf algebras of dimension 16 we also compute all the possible structures of the grothendieck ring of semisimple non commutative hopf algebras of dimension 16 moreover we prove that non commutative semisimple hopf algebras of dimension pn p prime cannot have a cyclic group of grouplikes

finite group factorizations and braiding we compute the quantum double braiding and other canonical hopf algebra constructions for the bicrossproduct hopf algebrahassociated to the factorization of a finite groupxinto two subgroups the representations of the quantum double are described by a notion of bicrossed bimodules generalising the cross modules of whitehead we also show that basis preserving self duality structures for the bicrossproduct hopf algebras are in one one correspondence with factor reversing group isomorphisms the example z6z6is given in detail we show further that the quantum doubled h is the twisting ofd x by a nontrivial quantum cocycle

the preference order of fuzzy numbers abstract many fuzzy number ranking approaches are developed in the literature for multiattribute decision making problems almost all of the existing approaches focus on quantity measurement of fuzzy numbers for ranking purpose in this paper we consider the ranking process to determine a decision maker s preference order of fuzzy numbers a new ranking index is proposed to not only take quantity measurement but incorporate quality factor into consideration for the need of general decision making problems for measuring quantity several cuts of fuzzy numbers are used a signal noise ratio is defined to evaluate quality of a fuzzy number this ratio considers the middle point and spread of each cut of fuzzy numbers as the signal and noise respectively a fuzzy number with the stronger signal and the weaker noise is considered better moreover the associated levels are treated as the degree of belief about the cut and used as weights in the index for strengthening the influence of cut with higher levels the membership functions of fuzzy numbers are not necessarily to be known beforehand while applying this index only a few left and right boundary values of cuts of fuzzy numbers are required we have proved the feature of the proposed index in a particular case several examples are also used to illustrate the feature and applicability in ranking fuzzy numbers

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

weakly group theoretical and solvable fusion categories we introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups weakly group theoretical categories and solvable categories these are fusion categories that are morita equivalent to iterated extensions in the world of fusion categories of arbitrary respectively solvable finite groups weakly group theoretical categories have integer dimension and all known fusion categories of integer dimension are weakly group theoretical our main results are that a weakly group theoretical category c has the strong frobenius property i e the dimension of any simple object in an indecomposable c module category divides the dimension of c and that any fusion category whose dimension has at most two prime divisors is solvable a categorical analog of burnside s theorem for finite groups this has powerful applications to classification of fusion categories and semsisimple hopf algebras of a given dimension in particular we show that any fusion category of integer dimension 84 is weakly group theoretical i e comes from finite group theory and give a full classification of semisimple hopf algebras of dimensions pqr and pq2 where p q r are distinct primes

on fuzzy number lattice r abstract we give a systematic development of fuzzy number lattice theory many of our results generalize to number lattice over the lattice algebra our first main result is that the real number chain r is a homomorphic image of the fuzzy number lattice r that is r r and the congruence class r is a component of r thus we can be specialized to describe the structure of the fuzzy number lattice r next we study the structure and representation of the congruence class

the unitary dual for complex classical lie groups in this paper we classify the unitary spectrum of the complex classical groups let g be any complex simply connected semisimple lie group there are two well known ways of constructing unitary representations from lower groups unitary induction from a parabolic subgroup and complementary series a reasonable way to describe the unitary dual is to find a distinguished set of representations for each group such that every other unitary representation is obtained from this set by the above constructions from strictly smaller parabolic subgroups this is entirely similar to the classification of admissible modules where the distinguished representations are the discrete series for integral infinitesimal character such a set is provided by the special unipotent representations introduced in bv1 their existence and unitarity was conjectured earlier by a they were shown to have many interesting properties such as the fact that their annihilator is a maximal primitive ideal from this it was possible to obtain character formulas for them however it is clear that this set is not large enough to describe the entire unitary spectrum for example in type c the metaplectic representation is not special unipotent or even one of the larger class of representations considered in a the definition of special unipotent representations suggests that one should consider representations with maximal primitive ideal and infinitesimal character as small as possible it is not clear which of them should be unitary imposing the condition that the annihilator be a completely prime ideal which is necessary for unitarity does not seem to be very helpful either thus one is left with the task of explicitly computing the g invariant form and ruling out the cases when it is indefinite this is an almost impossible calculation a rough estimate says that one needs to compute signatures on the k types in the principal series which have length ii 2p ql

ranking and defuzzification methods based on area compensation abstract we present some interesting properties related to the area compensation procedure to compare fuzzy numbers it has been proved that this method produces more than a fuzzy interval order it induces a ranking of fuzzy numbers some further results are given about the transitivity property and about computational aspects extensions to non normal fuzzy numbers and fuzzy quantities are also proposed

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 fuzzy numbers with index of optimism abstract fuzzy numbers are convenient for representing imprecise numerical quantities in a vague environment and their comparison or ranking is very important for application purposes despite many methods suggested in the literature there is no single measure that is universally applicable to a wide variety of situations this paper suggests a new method for comparing fuzzy numbers based on the combination of maximizing possibility and minimizing possibility using an index of optimism in 0 1 reflecting the decision maker s risk taking attitude the method is simple but has many comparative advantages

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

fuzzy nonlinear programming approach for evaluating and ranking process yields with imprecise data abstract process yield the percentage of processed product units passing inspection is a standard numerical measure of process performance in the manufacturing industry on the basis of the process yield expression an index s p k was developed to provide an exact measure of process yield for normally distributed processes most traditional studies measuring process capability are based on crisp estimates in which the output process measurements are precise however it is common that the measurements of process quality characteristics are insufficiently precise traditional approaches for evaluating process yield become unreliable in such cases therefore this study formulates fuzzy numbers to describe the quality characteristic measurements and applies two methods to construct the fuzzy estimation for s p k a nonlinear programming approach is provided to solve the level sets of the estimator and a testing procedure is presented for making decisions finally this concept is illustrated with an example and extended to solve the ranking problem of multiple yield indices

an approximate approach for ranking fuzzy numbers based on left and right dominance this study presents an approximate approach for ranking fuzzy numbers based on the left and right dominance the proposed approach only requires a few left and right spreads at some levels of fuzzy numbers to determine the respective dominance of one fuzzy number over the other the total dominance is then determined by combining the left and right dominance based on a decision maker s optimistic perspectives such a dominance is useful in ranking the fuzzy numbers when membership functions cannot be acquired the approach proposed herein is relatively simple in terms of computational efforts and is efficient when ranking a large quantity of fuzzy numbers by using a few left and right spreads two groups of examples demonstrate the accuracy and applicability of the proposed approach

unipotent representations attached to spherical nilpotent orbits a coadjoint nilpotent orbit inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of a complex semisimple lie group g is said to be spherical if it contains an open orbit of a borel subgroup we determine when and how to attach unitary representations to such an orbit for the real orthogonal and symplectic groups our results actually extend to a larger class of nilpotent coadjoint orbits

projectivity and freeness over comodule algebras let h be a hopf algebra and a an h simple right h comodule algebra it is shown that under certain hypotheses every h a hopf module is either projective or free as an a module and a is either a quasi frobenius or a semisimple ring as an application it is proved that every weakly finite in particular every finite dimensional hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras and the latter are frobenius algebras similar results are obtained for h simple h module algebras

transportation investment project selection using fuzzy multiobjective programming abstract this paper proposes the fuzzy multiobjective programming for the problem of transportation investment project selection tips the programming then uses the fuzzy spatial algorithm which calculates the performance of objective achievement and the requirement of resource utilization as of fuzziness under a complex and uncertain decision making environment there exists a certain degree of interdependence among these transportation investment projects this paper uses expert evaluation and conducts respectively with the consensus of most of the experts the decision on interdependency type complementary and substitutive and on the degree of fuzzy interdependence in every iteration of the fuzzy spatial algorithm the method of ranking fuzzy numbers must be used so as to obtain the ranking for selecting investment projects this paper has modified the method provided by kim and park in order that the preference of most of the decision makers or experts can be overlooked since the degree of optimism or pessimism can be demonstrated in the profitability of every investment project in the end this paper will employ a numerical example to illustrate the method forwarded

on the gan gross prasad conjecture for u p q in this paper we give a proof of the gan gross prasad conjecture for the discrete series of u p q there are three themes in this paper branching laws of a small a mathfrak q lambda branching laws of discrete series and inductive construction of discrete series these themes are linked together by a reciprocity law and the notion of invariant tensor product

finite quantum groups and quantum permutation groups abstract we give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups we also give examples of finite quantum groups which are not quantum permutation groups one such example occurs as a split abelian extension associated to the exact factorization s 4 z 4 s 3 and has dimension 24 we show that in fact this is the smallest possible dimension that a non quantum permutation group can have

calculations of some groups of hopf algebra extensions w let us recall the theory of group extensions with abelian kernels mc x chap iv let g be a group and m a g module the equivalence classes of group extensions m a e a g giving rise to the g module action on m form a group with respect to the baer sum furthermore this group is 2 isomorphic to the cohomology group h g m the theory is extended to the theory of hopf algebra extensions which w x are abelian in some sense h s we restrict our interests to hopf algebra extensions over a field k which are of the form

nilpotent orbits and complex dual pairs w x x w w w w x g end w and similarly for our main theorem describes the behaviour of closures of nilpotent orbits under the action of moment maps it is easy to see that for a nilpotent coadjoint orbit o g the set 1 o is the union of nilpotent coadjoint orbits in g it turns out that it is a closure of a single orbit theorem 1 1 let o g be a nilpotent coadjoint orbit there exists a unique nilpotent coadjoint orbit o g such that 1 o o

study on fuzzy optimization methods based on principal operation and inequity degree fuzzy optimization is a well known optimization problem in artificial intelligence manufacturing and management so establishing general and operable fuzzy optimization methods are important in both theory and application in this paper by distinguishing principal indices and secondary indices we give a method for comparing fuzzy information based on synthesizing effect and an operation for achieving fuzzy optimization based on a principal indices transformation further we propose an axiomatic system for fuzzy inequity degree based on the essence of constraint and give an instructive metric method for fuzzy inequity degree then by combining with genetic algorithm we give some fuzzy optimization methods based on principal operation and inequity degree denoted by bpo id fga for short finally we consider the convergence of our algorithm using the theory of markov chains and analyze its performance through two concrete examples all these indicate that bpo id fga can effectively merge decision preferences into the optimization process and that it also possesses better global convergence so it can be applied to many fuzzy optimization problems

comparison of clusters from fuzzy numbers fuzzy numbers are used for representation of numerical quantities in a vague environment their comparison or ranking is important for application purposes a new index for comparing of fuzzy numbers based on their geometrical properties is suggested in this paper this geometrical index is tested on a group of selected examples and compared with the other well known indexes a method for comparison of m tuples of fuzzy numbers and an algorithm for comparison of subsets clusters of similar closed m tuples of trapezoidal fuzzy numbers are presented

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

fuzzy mcdm based on ideal and anti ideal concepts this paper presents a novel fuzzy multiple criteria decision making mcdm based on the concepts of ideal and anti ideal points the concepts of fuzzy set theory and hierarchical structure analysis are used to develop a weighted suitability decision matrix to evaluate the weighted suitability of different alternatives versus various criteria the distance of different alternatives versus positive ideal solution and negative ideal solution are then obtained by using the proposed ranking method finally the relative approximation values of various alternatives versus positive ideal solution are ranked to determine the best alternative

an efficient approach to solving fuzzy madm problems abstract an approach using defuzzifying methods is proposed for the fuzzy multiple attribute decision making madm problem the computational effectiveness and efficiency of six defuzzifying methods combined with the simple additive weighting saw method are evaluated based on a comparison to the improved fuzzy weighted average ifwa followed by four fuzzy ranking methods numerical examples are discussed to demonstrate the implementation and effectiveness of the methods

functions on symmetric spaces and oscillator representation in this paper we study the l 2 functions on u 2n o 2n and mp n r we relate them using the oscillator representation we first study some isometries between various l 2 spaces using the compactification we defined in h he an analytic compactification of the symplectic group j differential geom 51 1999 375 399 these isometries were first introduced by betten and olafsson in f betten g olafsson causal compactification and hardy spaces for spaces of hermitian type pacific j math 200 2