technical importance ratings in fuzzy qfd by integrating fuzzy normalization and fuzzy weighted average fuzzy quality function deployment qfd has been extensively used for translating customer requirements crs into product design requirements drs in fuzzy environments existing approaches however for rating technical importance of drs in fuzzy environments are found problematic either incorrect or inappropriate this paper investigates how the technical importance of drs can be correctly rated in fuzzy environments a pair of nonlinear programming models and two equivalent pairs of linear programming models are developed respectively to rate the technical importance of drs the developed models are examined and illustrated with two numerical examples

metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

a comparison of discrete algorithms for fuzzy weighted average fuzzy weighted average fwa which can be applied to various fields such as engineering design decision analysis etc and as function of fuzzy numbers is suitable for the problem of multiple occurrences of fuzzy parameters additional fuzziness may be introduced in the alpha cut arithmetic this paper reviews and compares discrete algorithms for the fwas in both theoretical comparison and numerical comparisons as opposed to the linear programming algorithms that may be efficient but require the help of linear programming software an alternative efficient algorithm is also proposed the algorithm introduces an all candidate criteria ratings weights replaced benchmark adjusting procedure other than a binary dichotomy search in the existing methods the theoretical worst case comparison shows that the algorithms of guu and guh and our alternative algorithm require the same order of calculation complexity o n where n is the number of fwa terms yet the guh algorithm always requires 2 n 1 calculations to be performed the algorithm of guu requires the elemental comparison complexity o n and is superior to the other algorithms in the worst case on the other hand our alternative algorithm and lee and park algorithm have the same comparison complexity o n log n yet our alternative algorithm requires o n calculation complexity which is better than o n log n of the lee and park algorithm the numerical experiments show that the algorithm of guu and our alternative algorithm generally perform and converge faster by requiring fewer calculations than that of lee and park s algorithm the alternative algorithm requires a slightly smaller number of calculations than that of the guu algorithm due to the use of the convergent benchmark adjustment procedure rather than a somewhat fixed binary search conversely in the number of element comparisons lee and park s algorithm is shown numerically generally slightly better than the alternative algorithm due to the simple binary search scheme used the alternative algorithm also requires fewer comparisons than that of the guu algorithm furthermore in comparison of the average cpu time requirements the ranks of the results of these algorithms are consistent with those of the algorithms for the average numbers of calculations in general the alternative algorithm may provide an efficient alternative algorithm to the fwas in the worse case situation the guu algorithm still may be considered as an efficient alternative

a new approach to normalization of interval and fuzzy weights a new approach to normalization of interval and fuzzy weights based on the so called interval extended zero method is proposed the three desirable intuitively obvious properties of the normalization procedure are defined the main of them is based on the assumption that the sum of normalized interval or fuzzy weights should be an interval or a fuzzy value centered around 1 with a minimal width the advantages of a new approach are illustrated with the use of numerical examples it is shown that a new approach performs better than known methods for normalization of interval and fuzzy weights as it provides the results with the properties which are close to the desirable ones

harmonic maps and the schoen conjecture we show that every quasisymmetric homeomorphism of the circle h 2 admits a harmonic quasiconformal extension to the hyperbolic plane h 2 this proves the schoen conjecture

global convergence of karnik mendel algorithms karnik mendel km algorithms are the most commonly used iterative type reduction methods in interval type 2 fuzzy sets and systems as well as new techniques for computing the fuzzy weighted average fwa various extensions and improvements have been proposed however no proof has been provided for the convergence of these extensions it is necessary to provide the proof because many of the iterative algorithms may have divergence cases in the present study we provide a theoretical proof that km algorithms exhibit global convergence different initialization methods and iteration formats can always obtain the same unique optimal solution thus there are no concerns about the possibility of divergence in extensions of km algorithms our proof provides theoretical support for the applications of km algorithms especially the type reduction designs used in type 2 fuzzy systems and fwa computations because of the important roles of km algorithms in these methods

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

the fuzzy weighted average within a generalized means function the fuzzy weighted average is widely used to solve hierarchical evaluation problems including fuzzy consideration for the operations of scoring weighting and aggregating previous works considered the fuzziness of score and weight and used the additive function to aggregate these weighted scores this study considers the aggregation operator also as a fuzzy variable and uses a generalized means function to fuzzify the aggregation operator within a fuzzy weighted average in practice the proposed model not only considers both the relative important of the criteria and its achieved performance but also conveys the influence of the dm s decision maker s evaluation attitude thus the proposed model can flexibly reflect any dm s evaluation attitude such as open neutral or rigorous thereby the proposed model can make an objective evaluation that approaches a real decision making situation and thus has the potential to be a useful management tool for improved resolution of fuzzy hierarchical evaluation problems

vertex method for computing functions of fuzzy variables abstract this paper is to introduce a new approach the vertex method for computing functions of fuzzy variables the method is based on the cut concept and interval analysis the vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi occurrence of variables in the functional expression by conventional interval analysis methods the algorithm is very easy to implement and can be applied to many practical problems some examples are given to illustrate the applications

fuzzy arithmetic with requisite constraints abstract the purpose of this paper is to show that the standard fuzzy arithmetic does not take into account known constraints when applied to states of linguistic variables these constraints referred to as requisite constraints represent additional information when they are ignored as in the standard fuzzy arithmetic the obtained results exhibit in general information deficiency it is thus important as argued in the paper to revise fuzzy arithmetic to take relevant requisite constraints into account some typical requisite constraints are overviewed with a more extensive examination of the requisite equality constraint crisp or fuzzy

fuzzy analysis of statistical evidence bayesian classifiers are effective methods for pattern classification although their assumptions on the belief structure among attributes are not always justified in this paper we introduce a new classification method based on the possibility measure which does not require a precise belief model and in a sense it includes the bayesian classifiers as special cases this new classification method uses the fuzzy operators to aggregate attributes information evidence and it is referred to as fuzzy analysis of statistical evidence fase fase has several nice properties it is noise tolerant it can handle missing values with ease and it can extract statistical patterns from the data and represent them by knowledge of beliefs which in turn are propositions for an expert system thus from pattern classification to expert systems fase provides a linkage from inductive reasoning to deductive reasoning

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

methodology of linguistics evaluation in risk situations using fuzzy techniques abstract in this article the mathematical expression which determines the fuzzy weighted mean of a system in terms of a determined or specific configuration of estimates is studied its objective lies in obtaining fuzzy sets that are as close as possible to the natural expressions in systems which need an estimative calculation of risk whether it be of failure access or of any other type with incomplete and or unprecise data to do this the calculation fuzzy weighted mean of the fuzzy restriction associated with a system has been established as the proposed evaluation technique increasing the number of natural expressions and thus the number of fuzzy processes that are involved in the said calculation this increase has been achieved by analyzing all the possible results on the basis of the number of fuzzy sets in the numerator and denominator of the expression that denotes the fuzzy weighted mean of a system taking into account to what point of the calculation such an increase is feasible without impairing the resulting fuzzy set the importance of the resulting fuzzy set is the motivation behind effecting an enhancement process in this analytic phase since it determines a linguistic meaning that is taken as an element of comparison and reference for the obtention of the final natural expression the results of the proposed method have been verified by means of an interpreter which has been designed to deal with all the fuzzy operations which intervene in the different processes

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

a comparison of vertex method with jhe method the jhe method has been recently proposed as an approximation to the vertex method for solving the fuzzy weighted average problem in this study the usefulness of the jhe method is demonstrated via illustrative examples to be rather restricted and in practice cannot be regarded as an efficient alternative to the vertex method the appropriateness of the jhe method only appears at the situation where the number of criteria exceeds 11 and the exactness of the solution is not a primary concern

modeling uncertain variables of the weighted average operation by fuzzy vectors the paper deals with the fuzzy extension of the weighted average operation first we study the convenient ways how uncertain weights and weighted values can be modeled by fuzzy vectors we show that in comparison to a tuple of fuzzy numbers that have been used for modeling uncertain values of particular weights and weighted values up to now fuzzy vectors extend the possibilities of utilizing the vague expert information concerning the weights and weighted values next we focus on computation of a fuzzy weighted average of a fuzzy vector of weighted values with a fuzzy vector of weights we derive a general formula and we study its special forms the advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about its variables is involved in computation this fact is illustrated by several examples finally we briefly discuss the problem of defuzzification of the resulting fuzzy weighted average

an approximation to the extension principle using decomposition of fuzzy intervals this article presents a proposal for the decomposition of large ranges of uncertainty associated with the cartesian product of two fuzzy intervals we compute an approximation of the fuzzy set obtained by applying extension principle to a real function by means of this decomposition and piecewise linearization of the function it is proved the efficiency of the method via convergence in d metric we also extend the method to the case when the function has more than two arguments some examples are included to illustrate the proposed method and to compare it with others

forecasting methods using fuzzy concepts in this paper fuzzy concepts are applied in forecasting product price and sales in the semiconductor industry which is often conceived as a highly dynamic environment first two fuzzy forecasting methods including fuzzy interpolation fi and fuzzy linear regression flr are developed and discussed forecasts generated by these methods are fuzzy valued next the subjective beliefs about whether the industry is booming or slumping and the speed at which this change in prosperity takes place during a given period are also considered two subjective functions are defined and used to adjust fuzzy forecasts practically fuzzy forecasts are incorporated with fuzzy programming like fuzzy linear programming flp or fuzzy nonlinear programming fnp for mid term or long term planning advantages over traditional methods are shown in our examples

the extension principle and a decomposition of fuzzy sets we give an algorithm to decompose a fuzzy interval u using this decomposition and the multilinearization of a univariate function f we obtain an approximation of the fuzzy interval where is obtained from f by applying the extension principle we provide approximation bounds some numeric illustration is provided

percolation cluster numbers it has recently been suggested that there may be an infinite number of independent exponents hidden in the tails of the probability distribution of average percolation cluster numbers a simple approximation of non gaussian effects was used to deduce this result and we show that this approximation is questionable extensive simulations of the cluster distribution have been made and an interesting dependence of the cumulants on concentration and range of summation has been observed

simulation of discrete linear time invariant fuzzy dynamic systems in this paper a new approach to simulation of discrete linear time invariant dynamic systems with parameters and initial conditions described by fuzzy numbers is proposed this approach solves the problem of fuzzy simulation by determining at each iteration a nested family of intervals corresponding to different levels of certainty a cuts that enclose the possible system states using optimisation the originality of this approach consists in considering uncertain parameters as time invariant during the simulation process the optimisation problems associated with the approach proposed must be solved globally in order to guarantee that the minimum volume box enclosing the region of system states for each level of certainty has been derived then a global optimisation algorithm based on the theory of moments and linear matrix inequality relaxations suitable for the type of problems polynomial programs that appear in the fuzzy simulation algorithm is used finally two real engineering application examples allow assessing the performance of the approach proposed

type 1 owa operators for aggregating uncertain information with uncertain weights induced by type 2 linguistic quantifiers the owa operator proposed by yager has been widely used to aggregate experts opinions or preferences in human decision making yager s traditional owa operator focuses exclusively on the aggregation of crisp numbers however experts usually tend to express their opinions or preferences in a very natural way via linguistic terms these linguistic terms can be modelled or expressed by type 1 fuzzy sets in this paper we define a new type of owa operator the type 1 owa operator that works as an uncertain owa operator to aggregate type 1 fuzzy sets with type 1 fuzzy weights which can be used to aggregate the linguistic opinions or preferences in human decision making with linguistic weights the procedure for performing type 1 owa operations is analysed in order to identify the linguistic weights associated to the type 1 owa operator type 2 linguistic quantifiers are proposed the problem of how to derive linguistic weights used in type 1 owa aggregation given such type of quantifier is solved examples are provided to illustrate the proposed concepts

multi step ranking of alternatives in a multi criteria and multi expert decision making environment in this work we introduce a decision model in the form of a recursive aggregation algorithm that attempts to mimic a multi step ranking process of a set of alternatives in a multi criteria and multi expert decision making environment the main idea is rather intuitive each alternative is initially assigned a list of values representing the group experts opinion about the extent to which the alternative satisfies a set of given criteria then the values for each alternative are combined with the weighted mean operator according to vectors of weights one for each decision maker in the group these weights express the personal judgement of the decision makers about the relative importance of the individual criteria consequently a new vector of values is obtained for each alternative these new values are combined again with the weighted mean operator taking into account the different degrees of influence each decision maker accepts from the rest of the group the latter aggregation step is repeated again and again for each alternative until a consensus is attained

are there infinitely many independent exponents in the percolation cluster numbers abstract an infinite number of independent exponents is expected not in the average cluster numbers but in the tails of the their probability distribution this effect is somewhat similar to voltage distributions in random resistor networks

fuzzy weighted average an improved algorithm abstract this paper suggests a modification on the fuzzy weighted average fwa method developed by dong and wong 1987 a steepest descend ascend method is proposed to facilitate steps 3 and 4 of the fwa to find the minimum maximum of f x1 x2 xn w1 w2 wn i 1n wixi i 1n wi an example used by dong and wong is again employed to illustrate the new method the same results are obtained but the new method requires much less evaluations and computations the advantages and disadvantages of the two algorithms are discussed also several relevant theorems are stated and discussed in this paper and the proofs of these theorems are listed in the appendix

aggregation using the linguistic weighted average and interval type 2 fuzzy sets the focus of this paper is the linguistic weighted average lwa where the weights are always words modeled as interval type 2 fuzzy sets it2 fss and the attributes may also but do not have to be words modeled as it2 fss consequently the output of the lwa is an it2 fs the lwa can be viewed as a generalization of the fuzzy weighted average fwa where the type 1 fuzzy inputs are replaced by it2 fss this paper presents the theory algorithms and an application of the lwa it is shown that finding the lwa can be decomposed into finding two fwas since the lwa can model more uncertainties it should have wide applications in distributed and hierarchical decision making

on the normalization of interval and fuzzy weights the normalization of interval and fuzzy weights is often necessary in multiple criteria decision analysis mcda under uncertainty especially in analytic hierarchy process ahp with interval or fuzzy judgements the existing normalization methods based on interval arithmetic and fuzzy arithmetic are found flawed and need to be revised this paper presents the correct normalization methods for interval and fuzzy weights and offers relevant theorems in support of them numerical examples are examined to show the correctness of the proposed normalization methods and their differences from those existing normalization methods

uncertain robot environment modelling using fuzzy numbers abstract in this paper we present a fuzzy oriented methodology to model an uncertain geometric robot environment and to manipulate geometric uncertainties between robot coordinate frames we describe any geometric primitive of robot environment as a parameter vector in parameter space not only ill known values of the parameterized geometric primitives but the uncertain quantities of coordinate transformations are represented by means of fuzzy numbers restricted to appropriate membership functions for consistent interpretation about geometric primitives between different coordinate frames we manipulate these uncertain quantities using fuzzy arithmetic as an application we deal with a 2 dimensional straight line correspondence problem of a mobile robot vision system this example shows the usefulness of the proposed methodology

on various approaches to normalization of interval and fuzzy weights abstract the paper deals with the problem of fuzzification of the procedure of normalization of weights first the existing methods for normalization of interval and fuzzy weights are reviewed second we study the problem of normalization of a fuzzy vector of weights that expresses the joint possibility distribution of initial weights we show that a correct way is to apply the extension principle proposed by zadeh since the result of such normalization is the fuzzy vector of normalized weights that expresses the true joint possibility distribution of normalized weights further we establish some properties of this approach to normalization that are important from the point of view of real applications finally since an n tuple of non interactive interval or fuzzy weights can be viewed as a fuzzy vector of weights of a special kind we investigate normalization of such kind of fuzzy vectors of weights according to the extension principle we show that from the point of view of the way of modelling uncertain normalized weights the result of this approach can be directly compared only with the result of normalization proposed by wang and elhag 2006 35 we find out that it is not sufficient to express the result of normalization only by an n tuple of normalized interval or fuzzy weights together with the constraint that the sum of the weights is equal to 1 since it can cause a false increase of uncertainty in the model this fact is illustrated by an example

applying a fuzzy set based method for robust estimation of coupling loss factors finite element models have been used by many authors to provide accurate estimations of coupling loss factors although much progress has been achieved in this area little attention has been paid to the influence of uncertain parameters in the finite element model used to estimate these factors it is well known that in the mid frequency range uncertainty is a major issue in this context a spectral element method combined with a special implementation of a fuzzy set based method which is called the transformation method is proposed as an alternative to compute coupling loss factors the proposed technique is applied to a frame type junction which can consist of two beams connected at an arbitrary angle in this context two problems are investigated in the first one the influence of the confidence intervals of the coupling loss factors on the estimated energy envelopes assuming a unit power input is considered in the other problem the influence of the envelope of the input power obtained considering the confidence intervals of the coupling loss factors is also taken into account the estimates of the intervals are obtained by using the spectral element method combined with a fuzzy set based method results using a monte carlo analysis for the estimation of the coupling loss factors under the influence of uncertain parameters are shown for comparison and verification of the fuzzy method

rating technical attributes in fuzzy qfd by integrating fuzzy weighted average method and fuzzy expected value operator quality function deployment qfd is a planning and problem solving tool that is gaining acceptance for translating customer requirements into the technical attributes of a product deriving the rating order of technical attributes from input variables is a crucial step in applying qfd when the relative weights of customer requirements and the relationship measures between customer requirements and technical attributes are expressed as fuzzy numbers calculating the importance of each technical attribute falls into the category of fuzzy weighted average in which the derived membership function of the fuzzy importance of each technical attribute is not explicitly known thus most ranking methods are not suitable under these circumstances a method is proposed in this paper using fuzzy weighted average method in the fuzzy expected value operator in order to rank technical attributes in fuzzy qfd an example of a flexible manufacturing system design is cited to demonstrate the application of the proposed approach

harmonic maps between 3 dimensional hyperbolic spaces we prove that a quasiconformal map of the sphere s 2 admits a harmonic quasi isometric extension to the hyperbolic space h 3 thus confirming the well known schoen conjecture in dimension 3

fuzzy weighted average the linear programming approach via charnes and cooper s rule abstract to improve the method of dong and wong fuzzy sets and systems 21 1987 183 199 for obtaining the fuzzy weighted average a steepest descent ascent method was proposed by liou and wang fuzzy sets and systems 49 1992 307 315 in this paper we propose to replace the steepest descent ascent method by linear programming which is a much more powerful approach for large problems the reduction of the original nonlinear programming problem into a linear one is achieved by the use of the charnes and cooper s linear transformation method the example used by dong and wong is again used to illustrate the approach

quasi flats and rigidity in higher rank symmetric spaces in this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces such as sl n 2 n 3 theorem 8 1 is a major step towards the proof of quasiisometric rigidity of such lattices e we also give a different and effective proof of the theorem of kleiner leeb on the quasi isometric rigidity of higher rank symmetric spaces kl

analytical solution methods for the fuzzy weighted average for the fuzzy weighted average fwa despite various discrete solution algorithms and their improvements attempts at analytical solutions are very rare this paper provides an analytical solution method for the fwa based on the conclusions of the karnik mendel km algorithm compared with the two current popular kinds of a cut based computational methods for the fwa mathematical programming transformations and direct iterate computations our method is precise and has a concise structure efficient computation process and sound theoretical proofs we propose two algorithms for computing the analytical solution of the fwa two numerical examples illustrate our proposed approach

maximal surfaces and the universal teichmuller space we show that any element of the universal teichmuller space is realized by a unique minimal lagrangian diffeomorphism from the hyperbolic plane to itself the proof uses maximal surfaces in the 3 dimensional anti de sitter space we show that in adsn 1 any subset e of the boundary at infinity which is the boundary at infinity of a space like hypersurface bounds a maximal space like hypersurface in ads3 if e is the graph of a quasi symmetric homeomorphism then this maximal surface is unique and it has negative sectional curvature as a by product we find a simple characterization of quasi symmetric homeomorphisms of the circle in terms of 3 dimensional projective geometry

novel weighted averages versus normalized sums in computing with words in this paper some properties of novel weighted averages that are related to the concepts of possibility theory are examined it is shown that novel weighted averages have certain interpretations in terms of addition of interactive interval or fuzzy constraints to do this alternative forms of novel weighted averages are provided in particular an alternative form of novel weighted averages represented by the extension principle is determined it is shown that when fuzzy set models of words are obtained by collecting data from subjects in a computing with words setting interactive addition of fuzzy sets is not a well defined method and the optimization problems related to it may have no solutions although interactive addition is recommended in the literature for solving multicriteria decision making problems and for dealing with uncertain probabilities on the other hand novel weighted averages perform a specific normalization that guarantees that they always exist

on computing normalized interval type 2 fuzzy sets this paper explains how to compute normalized interval type 2 fuzzy sets in closed form and explains how the results reduce to well known results for type 1 fuzzy sets and interval sets such normalized interval type 2 fuzzy sets may be needed in linguistic probability computa tions or multiple criteria decision analysis under uncertainty index terms fuzzy weighted average fwa interval type 2 fuzzy sets it2 fss interval weighted average iwa linguistic probability linguistic weighted average lwa normalized interval type 2 fuzzy sets

super exponential convergence of the karnik mendel algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval t2 fs is an important operation in a type 2 fuzzy logic system where it is called type reduction but it is also a potentially time consuming operation the karnik mendel km iterative algorithms are widely used for doing this in this paper we prove that these algorithms converge monotonically and super exponentially fast both properties are highly desirable for iterative algorithms and explain why in practice the km algorithms have been observed to converge very fast thereby making them very practical to use

aggregation using the fuzzy weighted average as computed by the karnik mendel algorithms by connecting work from two different problems the fuzzy weighted average fwa and the generalized centroid of an interval type 2 fuzzy set a new alpha cut algorithm for solving the fwa problem has been obtained one that is monotonically and superexponentially convergent this new algorithm uses the karnik mendel km algorithms to compute the fwa cut end points it appears that the km cut algorithms approach for computing the fwa requires the fewest iterations to date and may therefore be the fastest available fwa algorithm to date

an efficient centroid type reduction strategy for general type 2 fuzzy logic system in this paper an efficient centroid type reduction strategy for general type 2 fuzzy set is introduced this strategy makes use of the result of a plane representation and performs the centroid type reduction on each a plane simulations show that it usually needs only several resolution of a value such that the defuzzified value converges to a real value consequently comparing with the exhaustive computation approach this approach can tremendously decrease the computation complexity from exponential into linear

first exit time of termites and random super normal conductor networks the resistance of a random super normal conductor network is calculated through the diffusion of a termite which performs a random walk on the normal bonds and has the same probability to exist from any point on a cluster of superconducting bonds the first time t that the termite exits at a distance r from the origin averaged over many configurations and runs is found to behave as t approximately r2k for r xi pc p s describes the divergence of the conductivity as p to pc p being the concentration of the superconductor scaling arguments are used to show that k 1 s 2 theta where 2 theta is the fractal dimension of random walks on single clusters at pc numerical simulations at two dimensions d 2 yields s 1 34 or 0 10 and k 1 34 or 0 03 in agreement with scaling the authors also show that the probability of return to the origin at time t behaves as t dk2 preliminary results at d 1 and other calculations methods are also discussed

alpha plane representation for type 2 fuzzy sets theory and applications this paper 1 reviews the alpha plane representation of a type 2 fuzzy set t2 fs which is a representation that is comparable to the alpha cut representation of a type 1 fs t1 fs and is useful for both theoretical and computational studies of and for t2 fss 2 proves that set theoretic operations for t2 fss can be computed using very simple alpha plane computations that are the set theoretic operations for interval t2 it2 fss 3 reviews how the centroid of a t2 fs can be computed using alpha plane computations that are also very simple because they can be performed using existing karnik mendel algorithms that are applied to each alpha plane 4 shows how many theoretically based geometrical properties can be obtained about the centroid even before the centroid is computed 5 provides examples that show that the mean value defuzzified value of the centroid can often be approximated by using the centroids of only 0 and 1 alpha planes of a t2 fs 6 examines a triangle quasi t2 fuzzy logic system q t2 fls whose secondary membership functions are triangles and for which all calculations use existing t1 or it2 fs mathematics and hence they may be a good next step in the hierarchy of flss from t1 to it2 to t2 and 7 compares t1 it2 and triangle q t2 flss to forecast noise corrupted measurements of a chaotic mackey glass time series

correspondence note on the lack of equality between fuzzy weighted average and fuzzy convex sum the paper shows that the equality of the interval valued weighted average based on interval weights and the convex sum based on normalized interval weights presented in dubois the role of fuzzy sets in decision sciences old techniques and new directions fuzzy sets syst 184 2011 3 28 is generally false for n 3 and discusses its consequences

fuzzy weighted average a max min paired elimination method by the use of max min paired elimination an efficient method for obtaining the fuzzy weighted average fwa was developed this approach is much more efficient than either the approach developed by dong and wong 1 or the improved approach developed by liou and wang 2 the computational requirements for the method of dong and wong are 22n and the largest computational requirements for the improved method of liou and wang are 2 n n 1 the computational requirements for the proposed approach are proportional to the problem size n and the total computational requirements are 2 n 1 the same example used by dong and wong is used to illustrate the approach in addition a more complicated problem is also solved

fuzzy system reliability analysis for components with different membership functions abstract this paper addresses the fuzzy system reliability analysis for components with different membership functions conventionally it is not always easy to obtain a system reliability for components with different individual failure probability density functions p d f due to the necessary but complicated combination and integration of various p d f s for such a problem and considering fuzzy environment and data imprecision in the real life we now apply fuzzy set theory to solve the inadequacy of the conventional probability in accounting and processing of built in uncertainties in the probabilistic data therefore we will use fuzzy distributions instead of the classical probability distribution for the components and calculate the functions of fuzzy numbers to solve for the fuzzy system reliability via non linear programming techniques where the commercial packages are available to use this solution procedure will be described in conjunction with an example used to illustrate the analysis algorithm and computation of this proposed method

fuzzy multiple attributes and multiple hierarchical decision making abstract a procedure is proposed to solve the multiple attributes and multiple hierarchical system under fuzzy environment the approach is based on 1 1 fuzzy representation 2 2 hierarchical performance evaluation structure 3 3 gradient eigenvector method for rating the fuzzy criteria weighting and 4 4 using the max min paired elimination method for aggregation to illustrate the approach an example on the evaluation of teaching performance in higher education is solved

uniqueness and regularity of proper harmonic maps introduction the purpose of this paper is to study some uniqueness and regularity properties of the dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces in general if the metrics of two complete noncompact manifolds mm and n are given locally by ds2 imj gij dxzdxj and dsy en h 3 du du respectively then the energy density function of a c1 map u m n is defined by

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we