combination of fuzzy numbers representing expert opinions abstract expert opinions or imprecise estimates of a physical variable are expressed as fuzzy numbers and five techniques for combining these numbers into a single fuzzy number estimate are developed seven characteristics of the combination technique are defined namely agreement preservation order independence transformation variance possibility conservation possibility interval conservation relationship between uncertainty of individual estimates and overall uncertainty and desirability of resultant estimate the five techniques listed in increasing order of preference are 1 crisp weighting 2 fuzzy weighting 3 minimal fuzzy extension 4 convex fuzzy extension and 5 mixed linear extension an example of estimating nitrate concentration in ground water illustrates the approach the cases of equally versus unequally reliable estimates are distinguished and guidelines for choice of combination technique are provided

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

the coordination index of finite collection of fuzzy sets this paper introduces a concept of the coordination index of finite collection of fuzzy sets the uniqueness of the representation of this index in metric lattice of fuzzy sets with continuous isotone estimation has been proved the correlation between isotone estimations of fuzzy set and its nearest usual set has been determined

fuzzy group decision making for evaluating the rate if aggregative risk in software development in this paper we present a new algorithm to evaluate the rate of aggregative risk in software development using fuzzy set theory under the fuzzy group decision making environment the proposed algorithm has the following advantages 1 it does not need to form the fuzzy assessment matrices for attributes 2 it does not need to perform the complicated defuzzification operations of fuzzy numbers using the centroid method the proposed algorithm is more efficient and faster in evaluating the rate of aggregative risk in software development under the fuzzy group decision making environment

a berndtsson andersson operator solving overline partial equation with wα estimates on convex domains of finite type the carleson measures were first introduced by carleson in order to solve the corona problem for the disc in mathbb c the notion of carleson measure can be generalized to any homogeneous space and were also used in the context of the corona problem in mathbb c n for example by varopoulos amar and andersson and carlsson one of the steps to solve the hp corona problem in a pseudoconvex domain is to solve the overline partial equation for a form satisfying a carleson condition and get norm estimates of the solution in term of the carleson norm of the main goal of this paper is to consider this question in the case of convex domains of finite type and to get estimates linked to the multitype of the domain

carleson embedding theorem on convex finite type domains abstract this paper concerns certain geometric aspects of function theory on smoothly bounded convex domains of finite type in c n specifically we prove the carleson hormander inequality for this class of domains and provide examples of carleson measures improving a known result concerning such measures associated to bounded holomorphic functions

fuzzy rules fuzzy functions a combination of logic and arithmetic operations for fuzzy numbers a general class of fuzzy numbers is introduced triangular fuzzy numbers tfns in extended representation this representation allows the operations of fuzzy arithmetic as well as fuzzy logic to be defined within this class of extended tfn at least approximately this leads naturally to the definition of fuzzy functions and it is shown that sugeno type fuzzy rules can be expressed by such functions in this context the sugeno controller is able to use fuzzy numbers as input variables thus extending the original concept of using crisp values as input

representation and application of fuzzy numbers abstract in this paper we present the theoretical background of fuzzy numbers connected with the possibility and dempster shafer theories we describe some types of representation of fuzzy numbers and we study the notions of the distance and orders between fuzzy numbers based on these representations the second part is devoted to the application of fuzzy numbers in data analysis artificial intelligence and decision making

two new approaches for assessing the weights of fuzzy opinions in group decision analysis the aggregation of fuzzy opinions is an important component of group decision analysis with fuzzy information this paper proposes two new approaches for the assessment of the weights to be associated with fuzzy opinions these approaches involve respectively the minimization of the sum of squared distances from one weighted fuzzy opinion to another which is called the least squares distance method lsdm and the minimization of the sum of squared differences between the defuzzified values of any two weighted fuzzy opinions which is called the defuzzification based least squares method dlsm the two approaches are developed and numerical examples are presented to illustrate their simplicity and effectiveness in aggregating fuzzy opinions

combining numerical and linguistic information in group decision making abstract people give information about their personal preferences in many different ways depending on their background this paper deals with group decision making problems in which the solution depends on information of a different nature i e assuming that the experts express their preferences with numerical or linguistic values the aim of this paper is to present a proposal for this problem we introduce a fusion operator for numerical and linguistic information this operator combines linguistic values assessed in the same label set with numerical ones assessed in the interval 0 1 it is based on two transformation methods between numerical and linguistic values which are defined using the concept of the characteristic values proposed in this paper its application to group decision making problems is illustrated by means of a particular fusion operator guided by fuzzy majority considering that the experts express their opinions by means of fuzzy or linguistic preference relations this operator is used to develop a choice process for the alternatives allowing solutions to be obtained in line with the majority of the experts opinions

choosing owa operator weights in the field of social choice one of the most important issues in the theory of owa operators is the determination of associated weights this matter is essential in order to use the best suited owa operator in each aggregation process given that some aggregation processes can be seen as extensions of majority rules to the field of gradual preferences it is possible to determine the owa operator weights by taking into account the class of majority rule that we want to obtain when individuals do not grade their pairwise preferences however a difficulty with this approach is that the same majority rule can be obtained through a wide variety of owa operators for this reason a model for selecting the best suited owa operators is proposed in this paper

a new algorithm for applying fuzzy set theory to evaluate the rate of aggregative risk in software development in this paper we present a new algorithm to tackle the rate of aggregative risk in fuzzy circumstances by fuzzy sets theory during any phase of the life cycle the proposed algorithm is easier more flexible and useful than the ones they have presented before

ck estimates for the equation on convex domains of finite type for a bounded convex domain open image in new window with c smooth boundary of finite type m and q 1 n 1 we construct a open image in new window solving integral operator t q such that for all k and the usual ck and open image in new window norms the operator open image in new window is continuous

zero varieties for the nevanlinna class in convex domains of finite type in mathbb c n with x ak each ak being counted according to its multiplicity in this case too it is well known that conversely the blaschke condition implies that the sequence x is the zero sequence of a function f e n q or even a bounded holomorphic function the question whether in dimension n 1 the blaschke condition b is sufficient for the divisor x xk nk to be defined by a function f e n q has been one of the main topics in this area the first result was independently obtained by henkin 11 and skoda 14 who proved it for strictly pseudoconvex domains with some obvious topological conditions also some partial results

a new method for tool steel materials selection under fuzzy environment abstract this paper presents a new method to solve the tool steel materials selection problem under fuzzy environment where the importance weights of different criteria and the ratings of various alternatives under different criteria are assessed in linguistic terms represented by fuzzy numbers because the proposed method uses simple arithmetic operations rather than the complicated arithmetic operations for aggregation and ranking of fuzzy numbers described in wang and chang 1995 its execution is much faster than the one presented in wang and chang 1995

aggregation of fuzzy opinions an axiomatic approach abstract the importance of fuzzy valuations in diagnostics is mentioned the problem of aggregation of fuzzy opinions obtained from a group of experts in answer to the question has an object the property labeled a and no property labeled 1a is formulated the set of fung and fu s 10 aggregation axioms is shortly described the idea of weighting expert s opinions is formalized then a new set of aggregation axioms is presented and an aggregating operator is introduced a posteriori weighting of opinions in aggregation is examined

on zero varieties of holomorphic functions in hardy spaces abstract in contrast to the famous henkin skoda theorem concerning the zero varieties of holomorphic functions in the nevanlinna class on the open unit ball bn in c n n 2 it is proved in this article that for any nonnegative increasing convex function t defined on r there exists g o b n satisfying s n g 1 d such that there is no f hp bn 0 z f z g here ng 1 denotes the integrated zero counting function associated with the slice function g this means that the zero sets of holomorphic functions belonging to the hardy spaces hp bn 0

zero sets of mathcal h p functions in convex domains of finite type we give a condition under which a divisor hat x in a bounded convex domain of finite type d in mathbb c n is the zero set of a function in a hardy space mathcal h p d for some p 0 this generalizes varopoulos result pac j math 88 189 246 1980 on zero sets of mathcal h p functions in strictly convex domains of mathbb c n

a new fuzzy multiple attributive group decision making methodology and its application to propulsion manoeuvring system selection problem in this paper a new fuzzy multiple attribute decision making fmadm method which is suitable for multiple attributive group decision making gdm problems in fuzzy environment is proposed to deal with the problem of ranking and selection of alternatives since the subjectivity imprecision and vagueness in the estimates of a performance rating enter into multiple attribute decision making madm problems fuzzy set theory provides a mathematical framework for modelling vagueness and imprecision in the proposed approach an attribute based aggregation technique for heterogeneous group of experts is employed and used for dealing with fuzzy opinion aggregation for the subjective attributes of the decision problem the propulsion manoeuvring system selection as a real case study is used to demonstrate the versatility and potential of the proposed method for solving fuzzy multiple attributive group decision making problems the proposed method is a generalised model which can be applied to great variety of practical problems encountered in the naval architecture from propulsion manoeuvring system selection to warship requirements definition

semiclassical limits for the hyperbolic plane we study the concentration properties of high energy eigenfunctions for the laplace beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions at the center of our investigation is the microlocal lift of an eigenfunction to sl 2 introduced by s zelditch the microlocal lift is based on s helgason s fourier transform and has a straightforward description in terms of the lie algebra sl 2 we begin with an elementary demonstration of zelditch s exact differential equation for the microlocal lift further for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity we show that the limit of microlocal lifts is a geodesic flow invariant positive measure our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the macdonald bessel functions we find that with a scaling of auxiliary parameters the corresponding high energy limit converges to the positive dirac measure for the lift to sl 2 of a single geodesic on the upper half plane in particular at high energy the microlocal lift of a macdonald bessel function is concentrated along the lift of a single geodesic we further find that the convergence is uniform in parameters and thus our considerations can be applied to study automorphic eigenfunctions in particular the maass cusp forms for certain cofinite subgroups of sl 2 a formula is presented for the microlocal lift of a cusp form as an integral of a family of dirac measures and a measure given directly by sums of products of the classical fourier coefficients of the cusp form the formula establishes that questions on quantum chaos for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients in particular we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums we extend the considerations to the case of the nonholomorphic eisenstein series for the modular group we find that a result of w luo and p sarnak is equivalent to a limit sum formula involving the riemann zeta function and the elementary divisor function the formula is suggestive of a formula of s ramanujan

estimates for the neumann problem for pseudoconvex domains in c2 of finite type abstract we outline results obtained for the neumann problem for an arbitrary pseudoconvex domain in c2 of finite type we obtain an approximation to the neumann operator a number of sharp estimates for the solution of u f are a consequence one of these is an extension of the l1 estimate of henkin and skoda used to characterize the zero sets of functions of the nevanlinna class

convex hypersurfaces and fourier transforms hilbert transforms and maximal functions along curves and surfaces spectral synthesis problems and the study of certain operators related to hyperbolic partial differential and pseudodifferential operators the problem of estimating such fourier transforms has a long history see for example hlawka 3 herz 2 littman 4 randol 9 10 svensson 15 sogge and stein 12 13 stein 14 marshall 5 etc the results of these investigations suggest a strong relation between the local curvature properties of the surface s and the decay of the function a we are mainly interested in how the geometry of s near a fixed point xo affects the decay of a6 for by using a partition of unity on s it is clearly enough to study integrals of the type

aggregation operators and fuzzy systems modeling abstract a general class of aggregation operators called mica having the properties of monotonicity symmetry and an identity element are introduced we stress the significance of the choice of identity in characterizing the operator we show that the t norm and t conorm are special cases of these operators other classees of these operators are introduced notably an additive class which is very much in the spirit of the kind of aggregation used in neural networks it is shown how a general description of the fuzzy system modeling technique can be obtained using these operators

on fourier coefficients of maass waveforms for psl 2 z in this paper we use machine experiments to test the validity of the sato tate conjecture for maass waveforms on psl 2 z h we also elaborate on stark s iterative method for calculating the fourier coefficients of such forms 1 introductory remarks around 20 years ago d h lehmer 19 empirically investigated the extent to which the numbers xp t p p 11 2 obey sato tate statistics as p oc here t n is the usual ramanujan tau function and the proposed statistics assert that 1 1 nlim j x op e 2 sin2odo xoo0 7r x 7f for xp 2cos op and any jordan measurable e c 0 7 7r x is the usual counting function for the primes the corresponding assertion for xp itself will then read 1 2 limn d 1 cf 33 37 18 and more recently 4 7 25 27 28 any exceptional cases should reduce to 10 received by the editor march 12 1992 and in revised form january 26 1993 1991 mathematics subject classification primary 11 f03 1 1 f30 the first author was partially supported by nsf grant dms 89 10744 with computer time from the minnesota supercomputer institute cray2 pittsburgh supercomputing center ymp832 and supercomputing research center cray2 lassociated with a z e27riz 17lo0 1 e27inz 24 1993 american mathematical society 0025 5718 93 1 00 25 per page

multicriteria decision analysis with fuzzy pairwise comparisons abstract we present a modification of the fuzzy multi criteria method proposed by van laarhoven and pedrycz 1983 first the weights of the decision criteria are calculated by the minimization of a logarithmic regression function next the weights of the decision alternatives are calculated for each criterion separately lastly the fuzzy final scores of the alternatives are determined by an appropriate aggregationm of the calculated weights we will show that our modified method in contrast with the original one yields weights which are optimal with respect to the logarithmic regression function we also present a version of our method with geometric ratio scales rather than saaty s equi distant scale to quantify the gradations of human comparative judgement this approach also enables us to develop an efficient procedure for investigating the scale sensitivity of the calculated weights and the final scores of the alternatives

a general multiple attribute decision making approach for integrating subjective preferences and objective information a general multiple attribute decision making madm approach is proposed to integrate subjective preferences and objective information and to assess the relative weights of attributes where subjective preferences are expressed either by a fuzzy preference relation on decision alternatives or by a pairwise comparison matrix on the relative weights of attributes or by both of them and objective information is expressed by a decision matrix three special cases of the approach are investigated which are the weighted least deviation norm wldn the weighted least square deviation norm wlsdn and the weighted minimax deviation norm wmdn approaches the extension of the general approach to group decision making and the choice of decision parameters are also discussed a numerical example is examined to show the applications of the proposed madm approaches

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

fuzzy group decision making for facility location selection the selection of a facility location among alternative locations is a multicriteria decision making problem including both quantitative and qualitative criteria the conventional approaches to facility location problem tend to be less effective in dealing with the imprecise or vagueness nature of the linguistic assessment under many situations the values of the qualitative criteria are often imprecisely defined for the decision makers the aim of the paper is to solve facility location problems using different solution approaches of fuzzy multi attribute group decision making the paper includes four different fuzzy multi attribute group decision making approaches the first one is a fuzzy model of group decision proposed by blin the second is the fuzzy synthetic evaluation the third is yager s weighted goals method and the last one is fuzzy analytic hierarchy process although four approaches have the same objective of selecting the best facility location alternative they come from different theoretic backgrounds and relate differently to the discipline of multi attribute group decision making these approaches are extended to select the best facility location alternative by taking into account quantitative and qualitative criteria a short comparative analysis among the approaches and a numeric example to each approach are given

weighted and boundary l p estimates for solutions of the equation on lineally convex domains of finite type and applications we obtain sharp weighted estimates for solutions of the equation u f in a lineally convex domain of finite type precisely we obtain estimates in the spaces l p being the distance to the boundary with two different types of hypothesis on the form f first if the data f belongs to l p 1 we have a mixed gain on the index p and the exponent secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic l p estimate with weight 1 moreover we extend those results to 1 and obtain l p and bmo estimates these results allow us to extend the l p regularity results for weighted bergman projection obtained in cdm14b for convex domains to more general weights

a note on the fuzzy weighted additive rule abstract the characteristic or attribute of a fuzzy weighted additive rule fwar is presented fwar has provided a method of assessing weights for criteria based on the fuzzy opinions of experts the method is to elicit weights for aggregating the fuzzy sets of type 1 and type 2 by using quadratic programming but fwar has not expressed a specific description on the information of experts overall evaluations for alternatives if the style of experts overall evaluations is characterized as linear additive weighting of the performances of alternative for criterion then the optimal aggregated value of the given performances of alternative is given by the arithmetic mean of the experts overall evaluations without solving time consuming quadratic programming in fwar

a method for fuzzy aggregation based on group expert evaluations this paper introduces an approach to group decision making based on the coordination index and similarity of the finite collections of fuzzy sets the mathematical apparatus of this approach based on the concept of maximal coordination is constructed in the metric lattice of fuzzy sets a specific kind of fuzzy aggregation operator is proposed a method for fuzzy aggregation is introduced thoroughly discussed and its algorithm determined the practical example of application of the introduced method is offered as well

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

asymptotic relations among fourier coefficients of automorphic eigenfunctions a detailed stationary phase analysis is presented for noncompact parameter ranges of the family of elementary eigenfunctions on the hyperbolic plane z y 1 2 k ir 2 my e 2 imx z x iy r 2 the eigenvalue s 2 m 1 2 and k ir the macdonald bessel function the phase velocity of on s imz 1 is a double valued vector field the tangent field to the pencil of geodesics g tangent to the horocycle s imz 1 for a sl 2 r a multi term stationary phase expansion is presented in a for az e 2 in rez uniform in parameters an application is made to find an asymptotic relation for the fourier coefficients of a family of automorphic eigenfunctions in particular for automorphic with coefficients an and eigenvalue a it is shown for the special range n 1 2 that a n is o 1 4 e 1 2 2 for a large improving by an order of magnitude for this special range upon the estimate from the general hecke bound o n 1 2 1 4 e 1 2 2 an exposition of the wkb asymptotics of the macdonald bessel functions is presented

on the uniform equidistribution of long closed horocycles it is well known that on any given hyperbolic surface of finite area a closed horocycle of length becomes asymptotically equidistributed as in this paper we prove that any subsegment of length greater than 1 2 e of such a closed horocycle also becomes equidistributed as the exponent 1 2 e is the best possible and improves upon a recent result by hejhal he3 we give two proofs of the above result our second proof leads to explicit information on the rate of convergence we also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to

aggregation of fuzzy opinions in a non homogeneous group an axiomatic treatment of an aggregation problem when the aggregated group is defined as a fuzzy group was proposed by the author in a previous paper in the present paper the fuzzy majority rule is characterized and an impossibility theorem for pure mixed rules is obtained moreover existence of absolutely decisive groups is studied

evaluating weapon systems using fuzzy arithmetic operations in this paper we present a new method to deal with the performance evaluation of weapon systems using fuzzy arithmetic operations an example of tactical missile systems selection is used to illustrate the performance evaluation process of weapon systems because the proposed methods uses simplified fuzzy arithmetic operations of fuzzy numbers rather than the complicated entropy weight calculations mentioned in 7 its execution is much faster than the one presented in 7

a method for aggregation of trapezoidal fuzzy estimates under group decision making this paper introduces a simple method for group decision making where experts opinions are expressed by trapezoidal fuzzy numbers the method is based on the metric approach providing an easy procedure for achieving a one valued consensus a new approach for determining experts degrees of importance is proposed a theoretical basis is established showing that the method is correct under the whole class of metrics on the set of trapezoidal fuzzy numbers a specific fuzzy aggregation operator is constructed for the realization of the proposed approach it is shown that despite its simplicity the proposed method preserves such important properties as agreement preservation order independence linearity property etc a comparative appraisal of the results obtained by the proposed approach and some other methods is given as well